Mandar S. Bhagwat,Ph.D.,April 12, 2005
NUCLEAR PHYSICS
ASPECTS OF NON-PERTURBATIVE QCD FOR HADRON PHYSICS (99 pp.)
Director of Dissertation:
Peter C. Tandy
Quenched lattice-QCD data on the dressed-quark Schwinger function can be correlated
with dressed-gluon data via a rainbow gap equation so long as that equation’s kernel
possesses enhancement at infrared momenta above that exhibited by the gluon alone.
The required enhancement can be ascribed to a dressing of the quark-gluon vertex. The
solutions of the rainbow gap equation exhibit dynamical chiral symmetry breaking and
are consistent with confinement. The gap equation and related, symmetry-preserving,
ladder Bethe-Salpeter equation yield estimates for chiral and physical pion observables
that suggest these quantities are materially underestimated in the quenched theory: |hq̄qi|
by a factor of two and fπ by 30 %.
Features of the dressed-quark-gluon vertex and their role in the gap and Bethe-Salpeter
equations are explored. It is argued that quenched lattice data indicate the existence of
net attraction in the colour-octet projection of the quark-antiquark scattering kernel.
The study employs a vertex model whose diagrammatic content is explicitly enumerable.
That enables the systematic construction of a vertex-consistent Bethe-Salpeter kernel
and thereby an exploration of the consequences for the strong interaction spectrum of
attraction in the colour-octet channel. With rising current-quark mass the rainbow-ladder
truncation is shown to provide an increasingly accurate estimate of a bound state’s mass.
Moreover, the calculated splitting between vector and pseudoscalar meson masses vanishes
as the current-quark mass increases, which argues for the mass of the pseudoscalar partner
of the Υ(1S) to be above 9.4 GeV.
A model for the dressed quark-gluon vertex, at zero gluon momentum, is formed
from a nonperturbative extension of the two Feynman diagrams that contribute at 1loop in perturbation theory. The required input is an existing ladder-rainbow model
Bethe-Salpeter kernel from an approach based on the Dyson-Schwinger equations; no new
parameters are introduced. The model includes an Ansatz for the triple-gluon vertex.
Two of the three vertex amplitudes from the model provide a point-wise description of the
recent quenched lattice-QCD data. An estimate of the effects of quenching is made. To be
phenomenologically tested this model has to be extended to non-zero gluon momentum.
Various options for such an extension are under consideration.
ASPECTS OF NON-PERTURBATIVE QCD FOR HADRON PHYSICS
A dissertation submitted to
Kent State University Department of Physics
in partial fulfillment of the requirements
for the degree of Doctor of Philosophy
by
Mandar S. Bhagwat
April 12, 2005
A dissertation written by
Mandar S. Bhagwat
B.Sc., Shivaji University, 1984
M.Sc., Shivaji University, 1986
Ph.D., Kent State University, 2004
Approved by
Dr. Peter Tandy , Chair, Doctoral Dissertation Committee
Dr. Richard Aron , Members, Doctoral Dissertation Committee
Dr. Alfred Cavaretta ,
Dr. Declan Keane ,
Accepted by
Dr. Makis Petratos , Chair, Department of Physics
Dr. John Stavley , Dean, College of Arts and Sciences
ii
Table of Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2 The Dyson-Schwinger Equation formalism . . . . . . . . . . . . . . . . . .
9
2.1
Dyson-Schwinger equations . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Gap Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3
n-point functions
2.4
Quark Dyson-Schwinger Equation . . . . . . . . . . . . . . . . . . . . . 14
2.5
9
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.1
Dressed-quark propagator . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.2
Dressed-gluon propagator
2.4.3
Dressed-quark-gluon vertex . . . . . . . . . . . . . . . . . . . . . . . 17
. . . . . . . . . . . . . . . . . . . . . . . . 16
Model for the quark DSE . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Quenched lattice-QCD dressed-quark propagator . . . . . . . . . . . . . . 24
3.1
Lattice gluon propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2
Effective quark-gluon vertex . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3
Fit to lattice results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4
Fidelity and quiddity of the procedure . . . . . . . . . . . . . . . . . . 28
3.5
Spectral properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.6
Chiral limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.7
Pion properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
iii
4 Algebraic model for the dressed quark-gluon vertex . . . . . . . . . . . . 41
4.1
Dressed quark-gluon vertex and the gap equation . . . . . . . . . . . 41
4.2
Vertex and interaction model . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.1
4.3
Algebraic vertex and gap equations . . . . . . . . . . . . . . . . . . . . 48
4.4
Algebraic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5
Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.5.1
5
Interaction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.6
Bethe-Salpeter Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.7
Bethe-Salpeter Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.7.1
Vertex consistent kernel . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.7.2
Solutions of the vertex-consistent meson Bethe-Salpeter equation . . 67
Quark-gluon vertex model and lattice-QCD data . . . . . . . . . . . . . . 77
5.1
One-loop perturbative vertex . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2
Nonperturbative vertex model . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3
Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6 Summary and Conclusions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
iv
List of Figures
1
Data, upper three sets: lattice results for M (p 2 ) in GeV at am values in Eq.
(3.12); lower points (boxes): linear extrapolation of lattice results [26] to
am = 0. Solid curves: best-fit-interaction gap equation solutions for M (p 2 )
obtained using the current-quark masses in Eq. (3.12); dashed-curve: gap
equation’s solution in the chiral limit, Eq. (2.50).
2
. . . . . . . . . . . . . . 28
Dimensionless vertex dressing factor: v(Q 2 ), defined via Eqs. (3.10), (3.11),
(3.13), obtained in the chiral limit (solid curve) and with the current-quark
masses in Eq. (3.12). v(Q2 ) is finite at Q2 = 0 and decreases with increasing
m(ζ).
3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Data, quenched lattice-QCD results for M (p 2 ) and Z(p2 ) obtained with
am = 0.036 [26]; dashed curve, Z(p2 ), and solid curve, M (p2 ) calculated
from the gap equation with our optimised effective interaction and m(ζ) =
55 MeV. (NB. Z(p2 ) is dimensionless and M (p2 ) is measured in GeV.) . . . 31
4
|∆S (T )| obtained from: the chiral limit gap equation solution calculated using our lattice-constrained kernel, solid curve; Eq. (3.18) with σ = 0.13 GeV,
θ = π/2.46, dotted curve; the model of Ref. [30], dashed curve. . . . . . . . 32
5
M (p2IR = 0.38 GeV 2 ), in GeV, as a function of the current-quark mass.
Solid curve, our result; circles, lattice data for am in Eq. (amvalues) [26];
dashed-line, linear fit to the lattice data, Eq. (3.27). . . . . . . . . . . . . . 35
v
6
Solid curve, calculated M (p2 = 0), in GeV, as a function of the currentquark mass m(ζ). The circles mark the current-quark masses in Eq. (3.12).
Dashed-line, linear interpolation of our result for M (p 2 = 0) on this mass
domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
7
Upper panel – C-dependence of A(s). For all curves m = 0.015. Solid line:
C = C¯ = 0.51; dash-dot-dot line: C = 1/4; dotted line: C = 0; and dash-dot
¯
curve: C = −1/8. Lower panel – Truncation-dependence of A(s), C = C.
Solid line: complete solution; dash-dash-dot line - result obtained with
only the i = 0, 1 terms retained in Eq. (4.24), the one-loop corrected vertex;
short-dash line - two-loop corrected; long-dash line - three-loop corrected;
and short-dash-dot line: four-loop corrected. In this and subsequent figures,
unless otherwise noted, dimensioned quantities are measured in units of G
in Eq. (4.21). A fit to meson observables requires G = 0.69 GeV and hence
m = 0.015 corresponds to 10 MeV. . . . . . . . . . . . . . . . . . . . . . . . 55
8
Upper panel – Current-quark-mass-dependence of the dressed-quark mass
function. For all curves C = C¯ = 0.51. Dotted line: m = m60 ; solid line:
m = 0.015; dashed line: chiral limit, m = 0. Lower panel – C-dependence
of M (s). For all curves m = 0.015. Solid line: C = C¯ = 0.51; dash-dot-dot
line: C = 1/4; dotted line: C = 0; and dash-dot curve: C = −1/8. In
addition, for C = 0.51: dash-dash-dot line - M (s) obtained with one-loop
corrected vertex; and short-dash line - with two-loop-corrected vertex. . . . 57
9
C-dependence of αC1 (s) in Eq. (4.22). For all curves m = 0.015. Solid line:
C = C¯ = 0.51; dash-dot-dot line: C = 1/4; dotted line: C = 0; and dash-dot
curve: C = −1/8. In addition, for C = 0.51: dash-dash-dot line - one-loop
corrected α1 (s); and short-dash line - two-loop-corrected result. . . . . . . . 60
vi
10
C-dependence of of αC2 (s). For all curves m = 0.015. Solid line: C = C¯ =
0.51; dash-dot-dot line: C = 1/4; and dash-dot curve: C = −1/8. Moreover,
for C = 0.51: dash-dash-dot line - one-loop result for α C2 (s); short-dash line
- two-loop result; long-dash line - three-loop; and short-dash-dot line: fourloop. For C = 0, αC2 (s) ≡ 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
11
Upper panel – Current-quark-mass-dependence of α C3 (s). For all curves
C = C¯ = 0.51. Dash-dot line: m = 2; dotted line: m = m 60 ; solid line:
m = 0.015; dashed line: chiral limit, m = 0. Lower panel – C-dependence
of αC3 (s). For all curves m = 0.015. Solid line: C = C¯ = 0.51; dash-dot-dot
line: C = 1/4; and dash-dot curve: C = −1/8. Moreover, for C = 0.51:
dash-dash-dot line - one-loop result for α C3 (s); short-dash line - two-loop
result; long-dash line - three-loop; and short-dash-dot line: four-loop. For
C = 0, αC3 (s) ≡ 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
12
¯ Eq. (4.61),
Upper panel – Impact of αC2 (s) and αC3 (s) on A(s). For C = C,
solid line: result obtained with both terms present; dashed-line: α C2 (s)
omitted; dash-dot line: αC3 (s) omitted. The dotted line is the result obtained
with both terms present in the vertex but C = −1/8. Lower panel – Impact
of αC2 (s) and αC3 (s) on M (s). In all cases m = 0.015. . . . . . . . . . . . . . 63
vii
13
Evolution of pseudoscalar and vector q q̄ meson masses with the currentquark mass. Solid line: pseudoscalar meson trajectory obtained with C =
C¯ = 0.51, Eq. (4.61), using the completely resummed dressed-quark-gluon
vertex in the gap equation and the vertex- consistent Bethe-Salpeter kernel;
short-dash line: this trajectory calculated in rainbow-ladder truncation.
Long-dash line: vector meson trajectory obtained with C¯ using the completely resummed vertex and the consistent Bethe-Salpeter kernel; dashdot line: rainbow-ladder truncation result for this trajectory. The dotted
vertical lines mark the current-quark masses in Table 3. . . . . . . . . . . . 72
14
Evolution with current-quark mass of the difference between the squaredmasses of vector and pseudoscalar mesons ( C¯ = 0.51) using the completely
resummed dressed-quark-gluon vertex in the gap equation and the vertexconsistent Bethe-Salpeter kernel. The dotted vertical lines mark the currentquark masses in Table 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
15
Evolution with current-quark mass of the relative difference between the
meson mass calculated in the rainbow-ladder truncation and the exact
value. Solid lines: vector meson trajectories; and dashed-lines; pseudoscalar
meson trajectories. The dotted vertical lines mark the current-quark masses
in Table 3. We used C¯ = 0.51. . . . . . . . . . . . . . . . . . . . . . . . . . . 75
16
The quark-gluon vertex at one loop. The left diagram labelled A is the
Abelian-like term ΓA
σ , and the right diagram labelled NA is the non-Abelian
term ΓNA
σ .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
viii
17
The amplitudes of the dressed quark-gluon vertex at zero gluon momentum and for quark current mass m(µ = 2 GeV) = 60 MeV. Quenched lattice data [37] is compared to the results of the DSE-Lat model [38]. The
Abelian Ansatz (Ward identity) is also shown except for λ 2 (p) which is
almost identical to the DSE-Lat model. . . . . . . . . . . . . . . . . . . . . 81
18
The amplitudes of the dressed quark-gluon vertex at zero gluon momentum,
and for quark current mass m(µ = 2 GeV) = 60 MeV, from two models:
DSE-Lat [38] and DSE-MT [30] that relate to quenched and unquenched
content respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
ix
List of Tables
1
Pion-related observables calculated using our lattice-constrained effective
interaction. m(ζ = 19 GeV) = 3.3 MeV was chosen to give m π = 0.1395 GeV.
The index “0” indicates a quantity obtained in the chiral limit.
2
. . . . . . 39
Calculated π and ρ meson masses, in GeV. (G = 0.69 GeV, in which case
m = 0.015 G = 10 MeV. In the notation of Ref. [40], this value of G corresponds to η = 1.39 GeV.) n is the number of loops retained in dressing
the quark-gluon vertex, see Eq. (4.24), and hence the order of the vertexconsistent Bethe-Salpeter kernel. NB. n = 0 corresponds to the rainbow√
ladder truncation, in which case mρ = 2 G, and that is why this column’s
results are independent of C. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3
Current-quark masses required to reproduce the experimental masses of
the vector mesons. The values of mηc , mηb are predictions. Experimentally
[100], mηc = 2.9797 ± 0.00015 and mηb = 9.30 ± 0.03. NB. 0−
ss̄ is a fictitious
pseudoscalar meson composed of unlike-flavour quarks with mass m s , which
is included for comparison with other nonperturbative studies. All masses
are listed in GeV.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
x
Chapter 1
Introduction
Interactions of quarks, leptons and gauge bosons are described by the Standard Model,
a theory of Electromagnetic, Weak and Strong forces.
Electrically charged particles interact by exchanging chargeless photons. The theory
of these electromagnetic interactions is Quantum Electrodynamics (QED). This means
that physical quantities such as the anamolous magnetic moment of the electron can be
calculated from first principles of QED. Also a state like positronium that does not appear
in the QED Lagrangian can be understood as a relativistic bound state of elementary
excitations of QED, where “elementary excitations” means those quantum fields that
appear in the Lagrangian of the theory. For QED these are electrons, positrons and
photons.
Strong interactions are due to the color charge carried by particles called quarks and
gluons, and the theory is accordingly named Quantum Chromodynamics (QCD). QED and
QCD are both gauge theories and so have structural similarities. But in detail they are
very different. The main difference is that because photons are electrically neutral they
cannot interact directly with each other, but gluons (the QCD equivalent of photons)
carry color charge and so interact directly with each other. It is an observational fact
that only colorless bound states of quarks and gluons, generically called hadrons, have
been observed. It is the direct interaction of colored gluons that seems responsible for
the non-existence of free colored particles. An understanding of color confinement was
listed as one of the top ten problems in particle physics for the millenium. Thus unlike
1
2
positronium which can be ionized with the addition of a few ( 7) electron-volts into an
electron and a positron, no amount of energy available presently is sufficient to break a
hadron into seperate colored constituents.
A central goal of contemporary nuclear physics is then to understand the properties
of hadrons in terms of the elementary excitations in QCD : quarks, gluons and ghosts.
More specifically hadron physics aims at solving the quantum field theoretical bound
state problem whose solution is the hadron spectrum; calculate the interactions of these
hadrons between themselves and with electroweak probes; and use these interactions as
incisive tools with which to identify the origins of confinement and elucidate its effects on
observables.
Again, because of the structural similarity between QED and QCD, and from the relatively straight forward calculations of QED observables from first principles, it is tempting
to assume that the problems in hadron physics could use the similar tools of perturbation
theory in QCD. But this is not true.
The strength of particle interactions is measured in terms of the coupling α =
g2
4π
where
g is the coupling constant. The use of the word ‘ ‘constant” is historical and a misnomer. α
varies(slowly) with the momentum of interaction and so the phrase “running coupling” is
the more pertinent. In QED α rises with increasing momentum (decreasing distance) and
αQED < 1 in the momentum range of 0 to few GeV. As a result perturbation theory yields
very accurate results, eg.1 part in 10 −10 for the electron’s anamolous magnetic moment.
In QCD α falls with increasing momentum (decreasing distance) and α s < 1 for momenta
& 2 GeV and perturbative QCD (pQCD) can be applied successfully for small distance
phenomena. But hadron physics requires solving QCD for momenta . 2 GeV where α s
is large and the problem becomes nonperturbative. This is the reason why calculations
based on QCD for observable particles are difficult: we know the theory to use, but do
3
not know how to solve it.
Much effort has been invested over the last thirty years in numerical simulations of
QCD on a finite spacetime grid. This approach to solving QCD, called lattice-QCD is now
an identifiable branch of high-energy physics with numerous large-scale collaborations,
not unlike the high-energy physics experimental collaborations; even the final results of
a lattice calculation are called lattice data. There are computational challenges faced by
lattice-QCD. Three of the principal ones are: reducing discretization errors (grid spacing and total lattice volume), extrapolating to small quark masses and going beyond the
quenched approximation. Significant efforts are being expended to develop improved algorithms so that discretization errors could be reduced. u and d quark masses are affected
most by dynamical chiral symmetry breaking (DCSB) effects in QCD, and the physics of
light quark mesons is dominated by these effects. As of today lattice has to extrapolate
to light quark masses and seeks help from other approaches developed to study hadron
physics. Quenched approximation introduces an a priori unquantifiable, systematic error.
In contrast to discretized lattice simulation, continuum tools in hadron physics depend
on modelling. Constituent-quark models describe the baryon spectrum and decays very
well using only a few parameters. But such models do not simultaneously give a satisfactory description of all the phenomena associated with light-quark mesons, which requires
an accurate description of DCSB. Light-front techniques are also useful in describing many
aspects of hadron physics as are effective field theories. Contemporary Dyson-Schwinger
equation (DSE) studies complement these approaches.
The DSEs are coupled integral (or differential) equations which relate the Green functions (i.e. Schwinger functions) of a field theory to each other. Solving these equations
provides a solution of the theory; QCD is completely defined when all of its n-point Green
functions are known, where n is the number of different external legs. The Bethe-Salpeter
4
equation (BSE) which describes the relativistic two-body scattering and bound states is
one of the DSEs. For studies based on DSEs it is inevitable that the infinite tower of
coupled equations must be truncated, i.e. the tower of equations must be limited to some
n. In this dissertation we shall discuss 2 and 3-point functions and the Bethe-Salpeter
equation. These truncations necessarily introduce a model for the omitted function. However, much can be achieved by maintaining certain properties of the theory, such as various
global and local symmetries, multiplicative renormalizability, known perturbative behavior in the weak coupling limit, etc. These can provide stringent conditions on the model.
In Chapter 2 the DSE formalism for QCD is introduced by deriving both the gap equation
for the quark propagator and the Bethe-Salpeter equation. The relevance of maintaining
the axial vector Ward-Takahashi identity is discussed. And arguments for using Euclidean
metric are provided.
It is a longstanding prediction from DSE studies that the Schwinger functions which
characterise the propagation of QCD’s elementary excitations: gluons, ghosts and quarks,
are strongly modified at infrared momentum scales, namely, spacelike momenta k 2 .
2 GeV2 [1, 2, 3]. Such momentum-dependent dressing is a fundamental feature of strong
QCD that is observable in hadronic phenomena [4]. For example: it is the mechanism by
which the current-quark mass evolves to assume the scale of a constituent-quark mass at
infrared momenta, and thereby the mechanism that exhibits dynamical chiral symmetry
breaking (DCSB); and it may also provide an understanding of confinement, as we discuss
in Chapter 3.
A great deal of progress in the QCD modeling of hadron physics has been achieved
through the use of the ladder-rainbow truncation of the Dyson-Schwinger equations (DSEs).
For two recent reviews, see Refs. [4] and [3]. Apart from 1-loop renormalization group
5
improvement, this truncation is built upon a bare quark-gluon vertex. Recent investigations with simple dressed vertex models have indicated that material contributions to a
number of observables are possible with a better understanding of the infrared structure of
the vertex. This is the central theme of the work presented in the following chapters. It is
certain that the infrared structure of the quark-gluon vertex, a 3-point function, has a big
impact on properties of the gap equation’s solution, such as: multiplicative renormalisability [5, 6, 7]; gauge covariance [8, 9, 6], and the existence and realisation of confinement
and DCSB [10, 11, 12, 13, 14, 15]. For example, related vertex Ansätze, which agree in the
ultraviolet, can yield solutions for the dressed-quark propagator with completely different
analytic properties and incompatible conclusions on DCSB [16, 17]. These diverse model
indications include an enhancement in the quark condensate [18, 19], an increase of about
300 MeV in the b1 /h1 axial vector meson mass [20], and about 200 MeV of attraction in
the ρ/ω vector meson mass.
Numerical simulations of lattice-QCD provide direct access to QCD’s Schwinger functions, and recent studies of the quenched theory yield dressed-gluon [21, 22, 23, 24] and
dressed -quark [25, 26] two-point functions (“propagators”) that are in semi-quantitative
agreement with earlier Dyson-Schwinger equation (DSE) calculations [27, 28, 29, 30, 31,
18, 32]. However, these dressed-gluon and -quark propagators are not obviously consistent
with each other in the following sense: use of the lattice dressed-gluon two-point function as the sole basis for the kernel of the QCDs gap equation cannot yield the lattice
dressed-quark propagator without a material infrared contribution (enhancement) from
the dressed-quark-gluon vertex [10, 11, 12]. Fortunately, just such behaviour can be understood to arise owing to multiplicative renormalisability of the gap equation [33, 34] and
is observed in lattice estimates of this three-point function [35, 36, 37].
In Chapter 3, we elucidate these points using a concrete model for the gap equation’s
6
kernel. Since the ultraviolet behaviour of this kernel is fixed by perturbative QCD and
hence model-independent, our study will focus on aspects of the infrared behaviour of the
Schwinger functions.
Furthermore, DCSB is encoded in the chiral-limit behavior of the dressed-quark propagator. However as mentioned above, contemporary lattice-QCD simulations are restricted
to current-quark masses that are too large for unambiguous statements to be made about
the magnitude of this effect. With a well-constrained model for the gap equation’s kernel
it is straightforward to calculate the dressed-quark propagator in the chiral limit. Hence
our analysis will also provide an informed estimate of the chiral limit behaviour of the
lattice results.
We consider two additional questions; namely, how do Schwinger functions obtained in
simulations of quenched lattice-QCD differ from those in full QCD, and can that difference
be used to estimate the effect of quenching on physical observables? Our model for the
gap equation’s kernel provides a foundation from which we believe these problems can
fruitfully be addressed.
As seen in Chapter 3 the gap equation study provides an understanding of the circumstances in which pointwise agreement is obtained [38, 39] with the lattice-data. This level
of sophistication does not prevail with the dressed-quark-gluon vertex, however. Acquiring
that is a realisable contemporary goal, and it is to aspects of this task that we address
ourselves Chapter 4 and 5.
Chapter 4 is concerned with an algebraic model for the dressed-quark-gluon vertex,
Γaν (q; p). We are generally interested in its form, how that arises, and the way it affects
strong interaction phenomena. After outlining some general properties of the dressedquark gluon vertex, we recapitulate on a nonperturbative DSE truncation scheme [40, 41]
7
that has already enabled some systematic study of the connection between the dressedquark-gluon vertex and the expression of symmetries in strong interaction observables. In
doing this we are led to propose an extension of earlier work, one which facilitates an exploration of the impact that aspects of the three-gluon vertex have on hadron phenomena.
To amplify the illustrative efficacy of our analysis we introduce a simple model to describe
the propagation of dressed-gluons [42] that reduces the relevant DSEs to a set of coupled algebraic equations which, notwithstanding their simplicity, exhibit characteristics
essential to the strong interaction.
We capitalise on the simplicity of our model and chronicle a range of qualitative features
of the dressed-quark-gluon vertex and dressed-quark propagator that are common to our
model and QCD. Of particular interest are the effects of net attraction in the colouroctet quark-antiquark scattering kernel which we are able to identify. We follow that
with an analysis of the Bethe-Salpeter equation which can be constructed, consistent with
the fully dressed-quark-gluon vertex, so that the Ward-Takahashi identities associated
with strong interaction observables are automatically satisfied. This property is crucial
to understanding hadron properties and interactions [43, 44, 46, 45, 47]. In addition, we
describe the evolution of pseudoscalar and vector meson masses with growing currentquark mass. One outcome of that is a quantitative assessment of the accuracy for meson
masses of the widely used rainbow-ladder truncation, which we determine by a comparison
with the masses obtained with all terms in the vertex and kernel retained.
After investigating the effects of the quark-gluon vertex with a simple model we attempt to provide a more realistic description of the vertex. In Chapter 5 we generate a
model dressed vertex, for zero gluon momentum, based on an Ansatz for non-perturbative
extensions of the only two diagrams that contribute at 1-loop order in perturbation theory.
An existing ladder-rainbow model kernel is the only required input. We compare to the
8
recent lattice-QCD data without parameter adjustment.
We first recall the vertex to 1-loop in perturbation theory and point out the structure
and properties that are used to suggest the Ansatz for non-perturbative extension. The
non-perturbative extension is described later and the results are presented and discussed.
Chapter 2
The Dyson-Schwinger Equation formalism
2.1
Dyson-Schwinger equations
The QCD action in Minkowski space of signature -2 can be written as,
S[ψ, ψ, A, φ, φ] =
Z
d4 x ψ(i∂/ − m0 + g0 A
/)ψ + L (A, φ, φ)
(2.1)
where ψ is the quark field, A = Aaµ ta is the gluon field and φ is the ghost field, t a , a = 1, ..., 8
are the SUc (3) generators in the fundamental representation and L (A, φ, φ) contains the
gluon and ghost terms. The action for N f quark flavors is a sum of the fermionic part of
S[ψ, ψ, A, φ, φ] over f with ψ → ψ f and m0 → mf0 .
The corresponding generating functional, including the sources is ,
Z[η, η, J , ω, ω] =
Z
D(ψ, ψ, A, φ, φ) exp(iS[ψ, ψ, A, φ, φ]+i
Z
d4 x (ψη+ηψ+A·J +ωφ+φω))
(2.2)
and because the integral of a derivative is zero [48], we get the following equation,
[(i∂/ − m0 + go γ µ
δ
δiJ µ (y)
)
δ
+ η(y)]Z = 0 ·
δiη(y)
(2.3)
This is the Dyson-Schwinger equation for the generating functional Z[η, η, J , ω, ω] [49].
Differentiating with respect to iη(x) yields a relation between the quark 2-point function
δ2 Z
δiη(x)δiη(y)
(i∂/ − m0 )
and the quark-gluon 3-point function
δ3 Z
δiη(x)δiη(y)iJµ (y) ,
δ2 Z
δ3 Z
δZ
+ g0 γ µ
= −iδ(x − y)Z + η(y)
δiη(x)δiη(y)
δiη(x)δiη(y)iJ µ (y)
δiη(x)
.
9
(2.4)
10
Higher derivatives relate higher n-point functions. These equations are called DSE for
n-point functions and equation (2.4) is often called the Dyson-Schwinger equation or the
gap equation. The main point to note is that calculation of the 2-point function requires
the knowledge of the 3-point function. This inter-dependence of 1, 2, ... , n, ... point
functions on each other is seen in all DSEs and they form an infinite number of coupled
equations, so that a nontrivial solution of even the simplest equation can only be obtained
after making truncations.
In the Dyson-Schwinger approach to hadron phenomenology the choice of truncation is
based on the preservation of the Ward-Takahashi identities. Also the formulation is carried
out in Euclidean space
1
. The motivation for implementing field theory in Euclidean
metric is briefly discussed later.
2.2
Gap Equation
After the sources are put to zero Eq(2.4) can schematically be written as,
ab
= i·
S0−1 iS + g02 ta γ µ SΓνb SDµν
(2.5)
Here Γµ is the fully amputated quark-gluon 3-point function also known as the fully dressed
quark-gluon vertex. Rearrangement gives,
ab
S −1 = S0−1 − ig02 ta γ µ SΓνb Dµν
(2.6)
which in momentum space takes the form
S −1 (p) = /p − m0 − Σ(p)
(2.7)
with the self-energy given by,
Σ(p) = i
1
Z
Λ
q
ab
g02 Dµν
(p − k)ta γµ S(k)Γbν (k, p).
We employ a Euclidean metric, with: {γµ , γν } = 2δµν ; 㵆 = γµ ; and a · b =
(2.8)
P4
i=1
ai bi .
11
This is the useful integral equation form of the gap equation. Here,
RΛ . RΛ 4
d q/(2π)4
q =
represents mnemonically a translationally-invariant regularisation of the integral, with Λ
the regularisation mass-scale. The final stage of any calculation is to remove the regularisation by taking the limit Λ → ∞.
After operating with
δ2
δiη(u)δiη(v)
on Eq(2.4), putting the sources to zero and then rear-
ranging the various terms gives the inhomogeneous Bethe-Salpeter equation for the 4-point
function S 12 (u, v, x, y) [48]. Symbolically it can be arranged as,
S 12 = S 1 S 2 − S 1 S 2 KS 12
(2.9)
where the kernel K is defined through the relation,
(i∂/ − m0 − Σ)(Σ + go ta γµ
δ
δiJ µa (y)
)S 12 =
Z
d4 x1 d4 x2 K(u, y; x1 , x2 )S(x1 , x2 ; v, z). (2.10)
In the vicinity of a bound state |Bi, S 12 can be written as [50]
S 12 =
P2
i|BihB|
+R
− M 2 + i
(2.11)
where R is regular in the vicinity P 2 = M 2 .
|Bi is of course an eigenstate of the QCD Hamiltonian but because we do not obtain
it as such, it cannot be normalized in the usual way. As a result its normalization condition must come from the inhomogeneous Bethe-Salpeter equation as P 2 → M 2 . The
inhomogeneous BSE can be rearranged as,
S 12 ((S 1 S 2 )−1 + K)S 12 = S 12 .
(2.12)
Substituting for S 12 from Eq(2.9) in this form of BSE we get the normalization condition
for the bound state,
hB|
∂
((S 1 S 2 )−1 + K)|Bi = −2iP µ
∂Pµ
(2.13)
12
where use was made of the homogeneous BSE for |Bi,
|Bi = S 1 S 2 K|Bi·
(2.14)
One more relation that is relevant to the present work is the axial vector-Ward Takahasi
Identity. This relation can be derived by observing the invariance of the 2-point function
under the following transformation,
ψ → ψ 0 = eiγ5 α(x) ψ
(2.15)
This function is invariant because the value of an integral does not change on changing
the variables and the measure can also be shown to be invariant in the flavor non-singlet
channel i.e. when the axial anomaly is not present. Thus we get,
∂µ h0|T (jµ5 (x)ψ(y)ψ(z))|0i − 2mh0|T (j 5 (x)ψ(y)ψ(z))|0i =
−h0|T (ψ(y)ψ(z))|0iδ(x − y) − h0|T (ψ(y)ψ(z))|0iδ(x − z)
(2.16)
where m is the renormalized quark mass and j 5 (x) = ψγ5 λα ψ and jµ5 (x) = ψγµ γ5 λα ψ
Since 2-legs can be extracted from h0|T (j µ5 (x)ψ(y)ψ(z))|0i and h0|T (j 5 (x)ψ(y)ψ(z))|0i
we are led to the amputated form of the AV-WTI identity which in momentum space is,
iPµ Γ5µ (P, k) = γ5 S −1 (k − P/2) + S −1 (k + P/2)γ5 − 2mΓ5 (P, k)
(2.17)
.
2.3
n-point functions
In Minkowski spacetime after a Pauli-Villars regularization, the time integral can be
evaluated either directly [51] or after Wick rotation[52]. In either case the positon and
nature of all singularities in the complex plane have to be known. In nonperturbative
studies these are not known because the singularities are dynamical and thus a part of
13
the final solution. Nonperturbative studies of DSEs in Euclidean space for QED and
QCD in the last 2 decades indicate that the singular behavior of 2-point functions of
electrons and quarks in the timelike region is quite different from the perturbative one,
due to the presence of complex conjugate poles or branch cuts. Thus the simple Minkowski
→ Euclidean Wick rotation that works so well in perturbation calculations is not firmly
established in nonpertubative formulations. n-point functions can be regularized and
renormalized in Euclidean theory as there are no dynamical singularities in the Euclidean
region. As a result one of the approaches to buiding field theoretic models is to start
in Euclidean space, obtain the regularized and renormalized n-point functions and then
analytically continue these functions into the timelike domain for calculating observables.
That this procedure is mathematically valid is discussed in [1] and references therein. .
The generating functional of Euclidean space QED or QCD is obtained from the one
in Minkowski space through the following transcription rules:
M inkowski → Euclidean,
Z
Z
d4 x → −i d4 x,
γ.∂ → −iγ.∂,
γ.A → −iγ.A,
A.B → −A.B
(2.18)
where the Minkowski metric has been chosen to have signature = -2. These rules follow
from the analytic continuation in the time variable x 0 → −ix4 with ~x → ~x
This method of analytic continuation of Schwinger functions is also used by the practitioners of lattice gauge theory allowing, as will be seen later, a pointwise comparison of
the n-point functions computed in these two independent approaches.
14
2.4
Quark Dyson-Schwinger Equation
The Euclidean space renormalised action is,
S[ψ, ψ, A, φ, φ] =
Z
d4 xψ(Z2 ∂/ + Z4 m + Z1 igA
/)ψ + L (A, φ, φ)
(2.19)
where Z1 (ζ, Λ), Z2 (ζ, Λ) and Z4 (ζ, Λ) are, respectively, Lagrangian renormalisation constants for the quark-gluon vertex, quark wave function and mass. m and g are the renormalized mass and renormalized coupling constant. Here, {γ µ , γν } = 2δµν , 㵆 = γµ and
P
a · b = 4i=1 ai bi . Then the DSE for the renormalised dressed-quark propagator can be
derived as before. It is
S(p)−1 = Z2 (iγ · p + m0 ) + Z1
Z
Λ
q
g 2 Dµν (p − q)
λa
γµ S(q)Γaν (q, p) ,
2
(2.20)
where Dµν (k) is the renormalised dressed-gluon propagator, Γ aν (q; p) is the renormalised
dressed-quark-gluon vertex, m0 is the Λ-dependent current-quark bare mass that appears
RΛ . RΛ 4
in the Lagrangian and q =
d q/(2π)4 represents mnemonically a translationallyinvariant regularisation of the integral, with Λ the regularisation mass-scale. The final
stage of any calculation is to remove the regularisation by taking the limit Λ → ∞. The
quark-gluon vertex and quark wave function renormalisation constants, Z 1 (µ2 , Λ2 ) and
Z2 (µ2 , Λ2 ) respectively, depend on the renormalisation point and the regularisation mass.
scale, as does the mass renormalisation constant Z m (µ2 , Λ2 ) = Z2 (µ2 , Λ2 )−1 Z4 (µ2 , Λ2 ). In
Eq. (2.20), S, Γaµ and m0 depend on the quark flavour, although we have not indicated this
explicitly. However, in our analysis we assume, and employ, a flavour-independent renormalisation scheme and hence all the renormalisation constants are flavour-independent.
2.4.1
Dressed-quark propagator
The solution of Eq. (2.20) has the general form
S(p)−1 = iγ · pA(p2 , µ2 ) + B(p2 , µ2 ) =
1
2
2
iγ
·
p
+
M
(p
,
µ
)
,
Z(p2 , µ2 )
(2.21)
15
renormalised such that at some large spacelike momenta µ
S(p)−1 p2 =µ2 = iγ · p + m(µ) ,
(2.22)
where m(µ) is the renormalised quark mass at the scale µ. In the presence of an explicit chiral symmetry breaking current-quark mass, one has Z 4 m(µ) = Z2 m0 , neglecting
O(1/µ2 ) corrections associated with dynamical chiral symmetry breaking that are intrinsically nonperturbative in origin.
Multiplicative renormalisability in QCD entails that
A(p2 , µ2 )
Z2 (µ2 , Λ2 )
1
=
= A(µ̄2 , µ2 ) =
.
2
2
2
2
2
A(p , µ̄ )
Z2 (µ̄ , Λ )
A(µ , µ̄2 )
(2.23)
Such relations can be used as constraints on model studies of Eq. (2.20). Explicitly, at
one-loop order in perturbation theory,
α(Λ2 )
Z2 (µ , Λ ) =
α(µ2 )
2
2
− γβF
1
,
(2.24)
where γF = 23 ξ and β1 = Nf /3 − 11/2, with ξ the gauge parameter and N f the number
of active quark flavours. At this order,
α(Q2 ) =
π
.
Q2
1
− 2 β1 ln Λ2
(2.25)
QCD
Clearly, at one-loop in Landau gauge [ξ = 0], A(p 2 , µ2 ) ≡ 1, and a deviation from this
result in a solution of Eq. (2.20) is a higher-loop effect. Such effects are always present in
the self-consistent solution of Eq. (2.20).
The ratio M (p2 , µ2 ) = B(p2 , µ2 )/A(p2 , µ2 ) is independent of the renormalisation point
in perturbation theory; i.e., with µ 6= µ̄,
.
M (p2 , µ2 ) = M (p2 , µ̄2 ) = M (p2 ) , ∀ p2 .
(2.26)
16
At one-loop order:
.
m(µ) = M (µ2 ) = m̂
1
2
ln
µ2
Λ2QCD
γm ,
(2.27)
where m̂ is a renormalisation-point-independent current-quark mass and γ m = 12/(33 −
2Nf ) is the anomalous dimension at this order; and
α(Λ2 )
Zm (µ , Λ ) =
α(µ2 )
2
2
γm
.
(2.28)
In QCD, γm is independent of the gauge parameter to all orders in perturbation theory
and the chiral limit is defined by m̂ = 0. Dynamical chiral symmetry breaking is manifest
when, for m̂ = 0, one obtains m(µ) ∼ O(1/µ 2 ) 6= 0 in solving Eq. (2.20), which is
impossible at any finite order in perturbation theory.
2.4.2
Dressed-gluon propagator
In a general covariant gauge the renormalised dressed-gluon propagator in Eq. (2.20)
has the general form
Dµν (k) =
δµν
kµ kν
− 2
k
kµ kν
d(k 2 , µ2 )
+ξ 4 ,
k2
k
(2.29)
where d(k 2 , µ2 ) = 1/[1 + Π(k 2 , µ2 )], with Π(k 2 , µ2 ) the renormalised gluon vacuum polarisation. The fact that the longitudinal [ξ-dependent] part of D µν (k) is not modified by
interactions is the result of a Slavnov-Taylor identity in QCD: k µ Dµν (k) = ξ kν /k 2 . We
note that Landau gauge is a fixed point of the renormalisation group; i.e., in Landau gauge
the renormalisation-group-invariant gauge parameter is zero to all orders in perturbation
theory; hence we employ this gauge in all numerical studies herein.
Multiplicative renormalisability entails that
d(k 2 , µ2 )
Z3 (µ̄2 , Λ2 )
1
=
= d(µ̄2 , µ2 ) =
.
2
2
2
2
2
d(k , µ̄ )
Z3 (µ , Λ )
d(µ , µ̄2 )
(2.30)
17
At one-loop order in perturbation theory
α(Λ2 )
Z3 (µ , Λ ) =
α(µ2 )
2
2
− βγ1
1
,
(2.31)
where γ1 = 13 Nf − 41 (13 − 3ξ).
2.4.3
Dressed-quark-gluon vertex
The renormalised dressed-quark-gluon vertex in Eq. (2.20) is of the form
Γaν (k, p) =
λa
Γν (k, p) .
2
(2.32)
As a fully amputated vertex, it is free of kinematic singularities. The general Lorentz
structure of Γν (k, p) is straightforward but lengthy, involving 12 distinct scalar form factors, and here we do not reproduce it fully:
Γν (k, p) = γν F1 (k, p, µ) + . . . ;
(2.33)
but remark that Ref. [53], pp. 80-83, and Refs. [95] provide an elucidation of its structure,
evaluation and properties.
Renormalisability entails that only the form factor F 1 , associated with the γν tensor,
.
is ultraviolet-divergent. By convention, and defining: f 1 (k 2 , µ2 ) = F1 (k, −k, µ) , Γν (k, p)
is renormalised such that at some large spacelike µ 2
f1 (µ2 , µ2 ) = 1 .
(2.34)
Since the renormalisation is multiplicative, one has
f1 (k 2 , µ2 )
Z1 (µ2 , Λ2 )
1
=
= f1 (µ̄2 , µ2 ) =
.
2
2
2
2
f1 (k , µ̄ )
Z1 (µ̄ , Λ )
f1 (µ2 , µ̄2 )
(2.35)
At one-loop in perturbation theory the vertex renormalisation constant is
α(Λ2 )
Z1 (µ2 , Λ2 ) =
α(µ2 )
where γΓ = 12 [ 34 (3 + ξ) + 34 ξ].
− γβΓ
1
,
(2.36)
18
2.5
Model for the quark DSE
As noted earlier, even the simplest DSE, the gap equation, can be solved only in an
approximate form. The approximation scheme used in our formulation is such that it
preserves the Axial Vector Ward Takashi Identity.
Consider the 2-flavor QCD Lagrangian in the chiral limit. Then it is invariant under
SUL (2) ⊗ SUR (2), besides other symmetries. As such, the currents with these symmetries
µ
jLµ and jR
, are conserved. The sum of these chiral currents corresponds to isospin currents
and its conservation to the conservation of isospin.
The difference is an axial vector current and does not correspond to any known conservation law of strong interactions. In 1960 Nambu and Jona-Lasinio hypothesized that
the symmetry corresponding to the conservation of this current is spontaneously broken.
Goldstone’s theorem states that every spontaneously broken continuous symmetry of a
quantum field theory leads to a massless particle with the quantum numbers of the broken symmetry generator. Thus in this case there would be a particle of negative parity,
zero spin, unit isospin and zero baryon number. The pion fits the description quite well,
except that it is not exactly massless. But its mass is much lower than other hadrons and
so can be blamed on the u and d quarks having a tiny mass. The question then is how to
incorporate this Goldstone boson character of the pion in our formulation?
We obtain the amplitudes for the mesons as solutions of the homogeneous BetheSalpeter equation (BSE), which has 2 inputs: 1) Solution of the quark propagator equation
and 2) Approximate form for the kernel K. To preserve the consequences of spontaneous
chiral symmetry breaking in this program, it has been shown in ref() that the truncation of
the gap equation and of the BSE must be consistent in the sense that the meson amplitudes
must satisfy the AVWTI.
19
To introduce an approximation, we use Eqs. (2.24), (2.31) and (2.36) and observe that
γ1
2γΓ
2γF
+
−
= 1.
β1
β1
β1
(2.37)
.
Hence, on the kinematic domain for which Q 2 = (p − q)2 ∼ p2 ∼ q 2 is large and spacelike, the renormalised dressed-ladder kernel in the Bethe-Salpeter equation for the (fullyamputated) Bethe-Salpeter amplitude behaves as follows:
2
2
g (µ ) Dµν (p − q)
= 4π α(Q
2
Γaµ (p+ , q+ )S
free
(p
) Dµν
(q+ ) ×
S(q− ) Γaν (q− , p− )
(2.38)
λa
λa
free
free
− q)
γµ S (q+ ) × S (q− ) γν ,
2
2
as can be seen ffrom the renormalised, homogeneous, pseudoscalar Bethe-Salpeter Equation (BSE)
[ΓH (k; P )]tu =
Z
q
Λ
rs
[χH (q; P )]sr Ktu
(q, k; P ) ,
(2.39)
.
where: H = π or K specifies the flavour-matrix structure of the amplitude; χ H (q; P ) =
S(q+ )ΓH (q; P )S(q− ), with S(q) = diag(Su (q), Sd (q), Ss (q)); q+ = q + ηP P , q− = q − (1 −
ηP ) P , with P the total momentum of the bound state; and r,. . . ,u represent colour-,
Dirac- and flavour-matrix indices.This observation, and the intimate relation between the
kernel of the pseudoscalar BSE and the integrand in Eq. (2.20) [40], provides a means
of understanding the origin of an often used Ansatz for D µν (k); i.e., in Landau gauge,
making the replacement
free
g 2 Dµν (k) → 4π α(k 2 ) Dµν
(k)
(2.40)
in Eq. (2.20), and using the “rainbow approximation”:
Γν (q, p) = γν .
(2.41)
The Ansatz expressed in Eq. (2.40) is often described as the “Abelian approximation”
because the left- and right-hand-sides are equal in QED. In QCD, equality between the
20
two sides of Eq. (2.40) cannot be obtained easily by a selective resummation of diagrams.
As reviewed in Ref. [1], Eqs. (5.1) to (5.8), it can only be achieved by enforcing equality
between the renormalisation constants for the ghost-gluon vertex and ghost wave function:
Z̃1 = Z̃3 .
A mutually consistent constraint, which follows from Z̃1 = Z̃3 at a formal level, is to
enforce the Abelian Ward identity Z 1 = Z2 . At one-loop this corresponds to neglecting
the contribution of the 3-gluon vertex to Γ ν , in which case γΓ → 32 ξ = γF . This additional
constraint provides the basis for extensions of Eq. (2.41); i.e., using Ansätze for Γ ν that
are consistent with the vector Ward-Takahashi identity in QED, such as Refs. [14, 17, 10].
The combination of Abelian and rainbow approximations [with Z 1 = 1 = Z2 ] yields
a mass function, M (p2 ), with the “correct” one-loop anomalous dimension; i.e., γ m in
Eq. (2.27) in the case of explicit chiral symmetry breaking or (1 − γ m ) in its absence [54].
However, other often used Ansätze for Γ ν [95, 5] yield different and incorrect anomalous
dimensions for M (p2 ) [55]. This illustrates and emphasises that the anomalous dimension
of the solution of Eq. (2.20) is sensitive to the details of the asymptotic behaviour of the
Ansätze for the elements in the integrand. One role of the multiplicative renormalisation
constant Z1 is to compensate for this.
An extensively studied model for the kernel of Eq. (2.20) is based on the Abelian
approximation [29, 30],:
Z1
Z
Λ
q
g 2 Dµν (p − q)
λa
γµ S(q)Γaν (q, p) →
2
Z
q
Λ
free
G((p − q)2 ) Dµν
(p − q)
λa
λa
γµ S(q) γν ,
2
2
(2.42)
with the specification of the model complete once a form is chosen for the “effective
coupling” G(k 2 ).
One consideration underlying this Ansatz is that while carrying out subtractive renormalisation in a DSE-model of QCD it is not possible to determine Z 1 without analysing
21
the DSE for the dressed-quark-gluon vertex. In [29] various Ansätze for Γ ν have been explored to show that, with G(k 2 ) = 4πα(k 2 ) for large-k 2 , there is always at least one Ansatz
for Z1 that leads to the correct anomalous dimension for M (p 2 ). This interplay between
the the renormalisation constant and the integral is manifest in QCD and Eq. (2.42) is a
simple means of implementing it.
Using Eqs. (2.20) and (2.42) our model quark DSE is
S(p, µ)−1 = Z2 iγ · p + Z4 m(µ) + Σ0 (p, Λ) ,
with the regularised quark self energy
Z Λ
λa
λa
.
0
free
Σ (p, Λ) =
G((p − q)2 ) Dµν
(p − q) γµ S(q) γν ,
2
2
q
(2.43)
(2.44)
Equation (2.43) is a pair of coupled integral equations for the functions A(p 2 , µ2 ) and
B(p2 , µ2 ) defined in Eq. (2.21).
In the case of explicit chiral symmetry breaking, m̂ 6= 0, the renormalisation boundary
condition of Eq. (2.22) is straightforward to implement. With
Eq. (2.22) entails
.
Σ0 (p, Λ) = iγ · p A0 (p2 , Λ2 ) − 1 + B 0 (p2 , Λ2 ) ,
Z2 (µ2 , Λ2 ) = 2 − A0 (µ2 , Λ2 ) and m(µ) = Z2 (µ2 , Λ2 ) m0 (Λ2 ) + B 0 (µ2 , Λ2 )
(2.45)
(2.46)
and hence
A(p2 , µ2 ) = 1 + A0 (p2 , Λ2 ) − A0 (µ2 , Λ2 ) ,
(2.47)
B(p2 , µ2 ) = m(µ) + B 0 (p2 , Λ2 ) − B 0 (µ2 , Λ2 ) .
(2.48)
From Sec. 2.4.1, having fixed the solutions at a single renormalisation point, µ, their
form at another point, µ̄, is given by
S −1 (p, µ̄) = iγ · p A(p2 , µ̄2 ) + B(p2 , µ̄2 ) =
Z2 (µ̄2 , Λ2 ) −1
S (p, µ) .
Z2 (µ2 , Λ2 )
(2.49)
22
[Recall that M (p2 ) is independent of the renormalisation point.] This feature is manifest
in our solutions. It means that, in evolving the renormalisation point to µ̄, the “1” in
Eq. (2.47) is replaced by Z2 (µ̄2 , Λ2 )/Z2 (µ̄2 , Λ2 ), and the “m(µ)” in Eq. (2.48) by m(µ̄);
i.e., the “seeds” in the integral equation evolve according to the QCD renormalisation
group.
As also remarked in Sec. 2.4.1, the chiral limit in QCD is unambiguously defined
by m̂ = 0. In this case there is no perturbative contribution to the scalar piece of the
quark self energy, B(p2 , µ2 ), and, in fact, there is no scalar, mass-like divergence in the
perturbative calculation of the self energy. It follows that
Z2 (µ2 , Λ2 )m0 (Λ2 ) = Z4 (µ2 , Λ2 )m(µ2 ) = 0 , ∀ Λ µ,
(2.50)
and, from Eqs. (2.46) and (2.48), that there is no subtraction in the equation for B(p 2 , µ2 );
i.e., Eq. (2.48) becomes
B(p2 , µ2 ) = B 0 (p2 , Λ2 ) ,
(2.51)
with limΛ→∞ B 0 (p2 , Λ2 ) < ∞.1 In terms of the renormalised current-quark mass the existence of DCSB means that, in the chiral limit, M (µ 2 ) ∼ O(1/µ2 ), up to ln µ2 -corrections.
An “Abelian approxiamtion” model for the “effective coupling” G(k 2 ) that has been
successful in describing physical properties of ground state pseudoscalar and vector mesons
is the Maris-Tandy (MT) model [30]. The Ansatz is
G(k 2 )
4π 2 D k 2 −k2 /ω2
=
e
+
k2
ω6
with γm =
12
33−2Nf
4π 2 γm F(k 2 )
2 ,
1
2
2
2 ln τ + 1 + k /ΛQCD
(2.52)
−s
and F(s) = (1 − exp( 4m
2 ))/s. The first term implements the strong
infrared enhancement in the region 0 <
t
k2
< 1 GeV 2 required for sufficient dynamical chi-
ral symmetry breaking. The second term serves to preserve the one-loop renormalization
1
This is a model-independent statement; i.e., it is true in any study that preserves at least the one-loop
renormalisation group behaviour of QCD.
23
group behavior of QCD. Here mt = 0.5 GeV, τ = e2 − 1, Nf = 4, and ΛQCD = 0.234 GeV.
The remaining parameters, ω = 0.4 GeV and D = 0.93 GeV 2 along with the quark masses,
are fitted to give a good description of hq̄qi, m π/K and fπ . The subsequent values for fK
and the masses and decay constants of the vector mesons ρ, φ, K ? are found to be within
10% of the experimental data [30].
Chapter 3
Quenched lattice-QCD dressed-quark propagator
Results from numerical simulations of lattice-QCD are available from recent studies
of the quenched theory for dressed-gluon [21, 22, 23, 24] and -quark [25, 26] two-point
functions (“propagators”). Use of the lattice dressed-gluon two-point function as the
sole basis for the kernel of QCD’s gap equation does not yield the lattice dressed-quark
propagator without a material infrared enhancement of the dressed-quark-gluon vertex
[10, 11, 12]. Here we elucidate this using a concrete model for the gap equation’s kernel.
Our analysis will also provide an informed estimate of the chiral limit behaviour of the
extrapolated lattice results.
3.1
Lattice gluon propagator
The dressed gluon propagator in Landau gauge (ζ = 0) is seen from Eq. (2.29)to have
the form,
Dµν (k) = Tµν (k) D(k 2 )
where D(k 2 ) =
d(k 2 ,µ2 )
k2
and Tµν = (δµν −
(3.1)
kµ kν
k2 )
In Ref. [21] the Landau gauge dressed gluon propagator was computed using quenched
lattice-QCD configurations and the result was parametrised as:
"
#
AΛ2α
L(k 2 , Λg )
g
2
D(k ) = Zg
+ 2
,
(k 2 + Λ2g )1+α
k + Λ2g
(3.2)
with
A = 9.8+0.1
−0.9 ,
α
=
2.2+0.1+0.2
−0.2−0.3 ,
Λg = 1.020 ± 0.1 ± 0.025 GeV,
Zg
=
24
2.01+0.04
−0.05 ,
(3.3)
25
where the first pair of errors are statistical and the second, when present, denote systematic
errors associated with finite lattice spacing and volume. In the simulation the lattice
spacing a = 1/[1.885 GeV]. The numerator in the second term of Eq. (3.2) is
−dD
1
1
1
2
2
2
L(k , Λg ) =
,
ln (k + Λg )( 2 + 2 )
2
k
Λg
(3.4)
with dD = [39 − 9 ξ − 4Nf ]/[2(33 − 2Nf )], an expression which ensures the parametrisation expresses the correct one-loop behaviour at ultraviolet momenta. In the quenched,
Landau-gauge study, Nf = 0 , ξ = 0, so
dD = 13/22 .
3.2
(3.5)
Effective quark-gluon vertex
To reiterate, asymptotic freedom entails that the “effective coupling” G(Q 2 ) is propor-
tional to the strong running coupling in the ultraviolet i.e.
G(k 2 ) = 4πα(k 2 ), Q2 & 2 GeV 2 ;
(3.6)
and so for Q2 & 2 GeV 2 (recall Eq. (2.25),
G(Q2 ) =
4π 2 γm
,
ln(Q2 /Λ2QCD )
(3.7)
where γm = 12/(33 − 2Nf ) is the anomalous mass dimension. To proceed we therefore
write
1
G(Q2 ) = D(Q2 ) Γ1 (Q2 ) ,
Q2
(3.8)
with D(Q2 ) given in Eq. (3.2) and
Γ1 (Q2 ) = 4π 2 γm
1 [ 21 ln(τ + Q2 /Λ2g )]dD
v(Q2 ) ,
Zg [ln(τ + Q2 /Λ2QCD )]
(3.9)
where τ = e2 −1 > 1 is an infrared cutoff. Equation (3.8) factorises the effective interaction
into a contribution from the lattice dressed-gluon propagator multiplied by a contribution
from the vertex, which we shall subsequently determine phenomenologically.
26
We remark that the renormalisation-group-improved rainbow truncation retains only
that single element of the dressed-quark-gluon vertex which is ultraviolet divergent at
one-loop level and this explains the simple form of Eq. (3.9). Systematic analyses of
corrections to the rainbow truncation show Γ 1 to be the dominant amplitude of the dressed
vertex: the remaining amplitudes do not significantly affect observables [41]. In proceeding
phenomenologically solely with Γ1 , we force v(Q2 ) to assume the role of the omitted
amplitudes to the maximum extent possible.
In Eq. (3.9), so long as v(Q2 ) ' 1 for Q2 & 2 GeV2 , Eq. (3.7) is satisfied and consequently the rainbow gap equation preserves the renormalisation group flow of QCD at
one-loop. We therefore consider a simple Ansatz with this property:
av (m) + Q2 /Λ2g
,
b + Q2 /Λ2g
(3.10)
a1
1 + a2 [m(ζ)/Λg ] + a3 [m(ζ)/Λg ]2
(3.11)
v(Q2 ) =
where
av (m) =
and a1,2,3 and b are dimensionless parameters, which are fitted by requiring that the gap
equation yield a solution for the dressed-quark propagator that agrees well pointwise with
the results obtained in numerical simulations of quenched lattice-QCD [25, 26]. It is
important to note that a good fit to lattice data is impossible unless a v (m) depends on
the current-quark mass. While more complicated forms are clearly possible, the Ansatz
of Eq. (3.11) is adequate.
3.3
Fit to lattice results
Now, to be explicit, the parameters in Eqs. (3.10), (3.11) were determined by the
following procedure. The rainbow gap equation; viz., Eq. (2.43) simplified via Eq. (2.44),
was solved using the effective interaction specified by Eqs. (3.8)–(3.11), with D(Q 2 ) exactly
as given in Eq. (3.2).
27
The ultraviolet behaviour of the mass function, M (p 2 ), is determined by perturbative
QCD and is therefore model independent. Hence the current-quark mass, m(ζ), was fixed
by requiring agreement between the DSE and lattice results for M (p 2 ) on p2 & 1 GeV2 .
We selected three lattice data sets from Ref. [26] and, for consistency with Refs. [29, 30],
used a renormalisation point ζ = 19 GeV, which is well into the perturbative domain. This
gave
a mlattice
0.018 0.036 0.072
.
(3.12)
m(ζ)(GeV) 0.030 0.055 0.110
The dimensionless parameters a1,2,3 and b were subsequently determined in a simultaneous least-squares fit of DSE solutions for M (p 2 ) at these current-quark masses to
all the lattice data. This necessarily required the gap equation to be solved repeatedly.
Nevertheless, the fit required only hours on a modern workstation, and yielded:
a1
a2
a3
1.5 7.35 63.0
b
.
(3.13)
0.005
These parameters completely determine the “best-fit effective-interaction” and hence our
lattice-constrained model for the gap equation’s kernel.
We emphasise that because the comparison is with simulations of quenched latticeQCD, we used Nf = 0 throughout and [35, 36, 37, 60]
Λqu−QCD = 0.234 GeV.
(3.14)
The strength of the running strong coupling is underestimated in simulations of
quenched lattice-QCD [61]. (NB. Halving or doubling Λ qu−QCD has no material quantitative impact on our results, nor does it qualitatively affect our conclusions.)
28
0.5
0.4
0.3
0.2
0.1
0.0
0.0
1.0
2.0
p (GeV)
3.0
4.0
Figure 1: Data, upper three sets: lattice results for M (p 2 ) in GeV at am values in Eq.
(3.12); lower points (boxes): linear extrapolation of lattice results [26] to am = 0. Solid
curves: best-fit-interaction gap equation solutions for M (p 2 ) obtained using the currentquark masses in Eq. (3.12); dashed-curve: gap equation’s solution in the chiral limit, Eq.
(2.50).
3.4
Fidelity and quiddity of the procedure
In Fig. 1 we compare DSE solutions for M (p 2 ), obtained using the optimised effective
interaction, with lattice results. In addition, we depict the DSE solution for M (p 2 ) calculated in the chiral limit along with the linear extrapolation of the lattice data to am = 0,
as described in Ref. [25]. It is apparent that the lattice-gluon and lattice-quark propagators can be correlated via the renormalisation-group-improved gap equation. That was
achieved via v(Q2 ) in Eq. (3.10), and the required form is depicted in Fig. 2. Plainly,
consistency between the propagators via this gap equation requires an infrared enhancement of the vertex, as anticipated in Refs. [10, 11, 33, 34]. Our inferred form is in semiquantitative agreement with the result of recent, exploratory lattice-QCD simulations of
the dressed-quark-gluon vertex [35, 36, 37].
Dynamical chiral symmetry breaking is another important feature evident in Fig. 1;
29
2.0
1.6
2
v(p )
1.8
1.4
1.2
1.0
0
2
4
6
2
2
p (GeV )
8
10
Figure 2: Dimensionless vertex dressing factor: v(Q 2 ), defined via Eqs. (3.10), (3.11),
(3.13), obtained in the chiral limit (solid curve) and with the current-quark masses in Eq.
(3.12). v(Q2 ) is finite at Q2 = 0 and decreases with increasing m(ζ).
viz., the existence of a M (p2 ) 6= 0 solution of the gap equation in the chiral limit. We
deduce that DCSB is manifest in quenched-QCD and, in the following, quantify the magnitude of that effect. It should be observed that a linear extrapolation to am = 0 of the
lattice data obtained with nonzero current-quark masses overestimates the mass function
calculated directly as the solution of the gap equation.
Figure 3 focuses on the lattice simulations for the intermediate value of the currentquark mass; namely, am = 0.036, and compares lattice output for both M (p 2 ) and the
quark wave function renormalisation, Z(p 2 ), with our results. We emphasise that the
form of Z(p2 ) was not used in fitting v(Q2 ). Hence the pointwise agreement between
the gap equation’s solution and the lattice result indicates that our simple expression for
the effective interaction captures the dominant dynamical content and, in particular, that
omitting the subdominant amplitudes in the dressed-quark-gluon vertex is not a serious
flaw in this study.
30
3.5
Spectral properties
In a quantum field theory defined by a Euclidean measure [62] the Osterwalder-
Schrader axioms [63, 64] are five conditions which any moment of this measure (n-point
Schwinger function) must satisfy if it is to have an analytic continuation to Minkowski
space and hence an association with observable quantities. One of these is “OS3”: the
axiom of reflection positivity, which is violated if the Schwinger function’s Fourier transform to configuration space is not positive definite. The space of observable asymptotic
states is spanned by eigenvectors of the theory’s infrared Hamiltonian and no Schwinger
function that breaches OS3 has a correspondent in this space. Consequently, the violation
of OS3 is a sufficient condition for confinement.
This connection has long been of interest [65, 42], and is discussed at length in Refs.
[66, 67, 68], and reviewed in Sec. 6.2 of Ref. [1], Sec. 2.2 of Ref. [2] and Sec. 2.4 of
Ref. [3]. It suggests and admits a practical test [10] that has been exploited in Refs.
[18, 10, 11, 69, 70, 71, 72, 73, 74, 75] and which for the quark 2-point function is based on
the behaviour of
∆S (T ) =
=
Z
1
π
3
d x
Z ∞
Z
d4 p ip·x
e
σS (p2 )
(2π)4
dε cos(ε T ) σS (ε2 ) ,
(3.15)
(3.16)
0
where σS is the Dirac-scalar projection of the dressed-quark propagator. For a noninteracting fermion with mass µ,
∆free
S (T )
1
=
π
Z
∞
dε cos(ε T )
0
1
µ
= e−µT .
ε2 + µ 2
2
(3.17)
The r.h.s. is positive definite. It is also plainly related via analytic continuation (T → it)
to the free-particle solution of the Minkowski space Dirac equation. The existence of an
associated asymptotic state is indubitable.
31
1.0
0.8
0.6
0.4
0.2
0.0
0.0
1.0
2.0
p (GeV)
3.0
4.0
Figure 3: Data, quenched lattice-QCD results for M (p 2 ) and Z(p2 ) obtained with am =
0.036 [26]; dashed curve, Z(p2 ), and solid curve, M (p2 ) calculated from the gap equation
with our optimised effective interaction and m(ζ) = 55 MeV. (NB. Z(p 2 ) is dimensionless
and M (p2 ) is measured in GeV.)
If instead one encountered a theory in which [67, 68, 75]
µ
1
1
+
,
σS (p ) =
2 p2 + µ2 − iρ2 p2 + µ2 + iρ2
2
(3.18)
a function with a pair of complex conjugate poles at p 2 + σ 2 exp(±iθ) = 0, where
σ 4 = µ4 + ρ4 , tan θ = ρ2 /µ2 ,
(3.19)
then
∆S (T ) =
µ −σT cos θ
θ θ
2 cos
.
e
σT sin +
2σ
2 2
(3.20)
This Fourier transform has infinitely many, regularly spaced zeros and hence OS3 is violated. Thus the fermion described by this Schwinger function has no correspondent in the
space of observable asymptotic states.
It is readily apparent that Eq. (3.18) evolves to a free particle propagator when ρ → 0.
This limit is expressed in Eq. (3.20) via θ → 0, σ → µ, wherewith Eq. (3.17) is recovered;
32
0
10
-1
10
-2
10
10
-3
-4
10
10
-5
-6
10
0
5
10
15
20
-1
25
30
T (GeV )
Figure 4: |∆S (T )| obtained from: the chiral limit gap equation solution calculated using
our lattice-constrained kernel, solid curve; Eq. (3.18) with σ = 0.13 GeV, θ = π/2.46,
dotted curve; the model of Ref. [30], dashed curve.
a result that is tied to the feature that the first zero of ∆ S (T ) in Eq. (3.20) occurs at
z1 =
π−θ
θ
csc
2σ
2
(3.21)
and hence z1 → ∞ for ρ → 0 [69, 70].
We calculated ∆S (T ) using the gap equation solutions discussed above and the form of
|∆S (T )| obtained with the chiral limit solution is depicted in Fig. 4. The violation of OS3
is manifest in the appearance of cusps: ln |∆ S (T )| is negative and infinite in magnitude
at zeros of ∆S (T ). The differences evident in a comparison with the result obtained from
Eq. (3.20) indicate that the singularity structure of the dressed-quark 2-point Schwinger
function obtained from the lattice-constrained kernel is more complicated than just a single
pair of complex conjugate poles. However, the similarities suggest that this picture may
serve well as an idealisation [75]. (NB. Qualitatively identical results are obtained using
σV (p2 ), the Dirac-vector projection of the dressed-quark propagator, instead of σ S (p2 ).)
33
Figure 4 also portrays the result obtained with the effective interaction proposed in
Ref. [30] and used efficaciously in studies of meson properties [4]. Significantly, the first
zero appears at a smaller value of T in this case. Using the model of Eq. (3.18) as a guide,
that shift indicates a larger value of σ. This sits well with the fact that the mass-scale
generated dynamically by the interaction of Ref. [30] is larger than that produced by the
interaction used herein, which is only required to correlate quenched lattice data for the
gluon and quark two-point functions.
From the results and analysis reported in this subsection, we deduce that light-quarks
do not appear in the space of observable asymptotic states associated with quenched-QCD;
an outcome anticipated in Ref. [34].
3.6
Chiral limit
The scale of DCSB is measured by the value of the renormalisation-point dependent
vacuum quark condensate, which is obtained directly from the chiral limit dressed-quark
propagator [43, 29]:
−hq̄qi0ζ
= lim Z4 (ζ, Λ) Nc tr
Λ→∞
Z
Λ
S0 (q),
(3.22)
q
where “tr” denotes a trace only over Dirac indices and the index “0” labels a quantity
calculated in the chiral limit, Eq. (2.50). In Eq. (3.22) the gauge parameter dependence
of the renormalisation constant Z4 is precisely that required to ensure the vacuum quark
condensate is gauge independent. This constant is fixed by the renormalisation condition,
Eq. (2.22), which entails
Z4 (ζ, Λ) = −
1 1
tr Σ0 (ζ, Λ)
m(ζ) 4
(3.23)
wherein the right hand side is well-defined in the chiral limit [29]. (It may also be determined by studying the fully amputated pseudoscalar-quark-antiquark 3-point function
34
[43].) A straightforward calculation using our chiral limit result; i.e, the propagator corresponding to the dashed-curve in Fig. 1, yields
−hq̄qi0ζ=19
GeV
= (0.22 GeV)3 .
(3.24)
To evolve the condensate to a “typical hadronic scale,” e.g., ζ = 1 GeV, one may use
[84]
hq̄qi0ζ 0 = Z4 (ζ 0 , ζ)Z2−1 (ζ 0 , ζ) hq̄qi0ζ =: Zm (ζ 0 , ζ)hq̄qi0ζ ,
(3.25)
where Zm is the gauge invariant mass renormalisation constant. Contemporary phenomenological approaches employ the one-loop expression for Z m and following this expedient we obtain, practically as a matter of definition,
ln[1/Λqu−QCD ] γm
(−hq̄qi019 GeV )
−hq̄qi01 GeV =
ln[19/Λqu−QCD ]
= (0.19 GeV)2 ,
(3.26)
which may be compared with a best-fit phenomenological value [76]: (0.24 ± 0.01 GeV) 3 .
It is notable that DSE models which efficaciously describe light-meson physics; e.g., Refs.
[29, 30], give hq̄qi01 GeV = −(0.24 GeV)3 .
Our gap equation assisted estimate therefore indicates that the chiral condensate in
quenched-QCD is a factor of two smaller than that which is obtained from analyses of
strong interaction observables. These results are in quantitative agreement with Ref. [34].
A fit to the linear extrapolation of the lattice data; viz., to the boxes in Fig. 1, gives
a significantly larger value [26]: −hq̄qi 01 GeV = (0.270 ± 0.027 GeV)3 . However, the error is
purely statistical. The systematic error, to which the linear extrapolation must contribute,
was not estimated. That may be important given the discrepancy, conspicuous in Fig. 1,
between our direct evaluation of the chiral limit mass function and the linear extrapolation
of the lattice data to am = 0: the linear extrapolation lies well above the result of our
chiral limit calculation.
35
0.400
0.300
0.200
0.100
0.000
0.025
0.100
0.050
0.075
m(ζ=19 GeV) (GeV)
0.125
Figure 5: M (p2IR = 0.38 GeV 2 ), in GeV, as a function of the current-quark mass. Solid
curve, our result; circles, lattice data for am in Eq. (amvalues) [26]; dashed-line, linear fit
to the lattice data, Eq. (3.27).
This point may be illustrated further. In Fig. 5 we plot M (p 2 = p2IR ), where pIR =
0.62 GeV, as a function of the current-quark mass (solid curve). This value of the argument
was chosen because it is the smallest p 2 for which there are two lattice results for M (p 2 )
at each current-quark mass in Eq. (3.12). Those results are also plotted in the figure.
It is evident that on the domain of current-quark masses directly accessible in lattice
simulations, the lattice and DSE results lie on the same linear trajectory, of which
M (p2IR = 0.38 GeV 2 , m(ζ)) = 0.18 + 1.42 m(ζ)
(3.27)
provides an adequate interpolation. However, as apparent in the figure, this fit on am ∈
[0.018, 0.072] provides a poor extrapolation to m(ζ) = 0, giving a result 40% too large.
In Fig. 6 we repeat this procedure, focusing solely on our value of M (p 2 = 0) because
directly calculated lattice data are unavailable at this extreme infrared point and published estimates obtained by extrapolating functions fitted to the lattice-p 2 -dependence
36
are inconsistent [25]. The pattern observed in Fig. 5 is again visible. On the domain of
current-quark masses for which lattice data are available, the mass-dependence of M (p 2 )
is well-approximated by a straight line; namely,
M (p2 = 0, m(ζ)) = 0.37 + 0.68 m(ζ) ,
(3.28)
but the value of M (0) determined via extrapolation to m(ζ) = 0 is 14% too large.
In all cases our calculated result possesses significant curvature.
At fixed p 2 ,
M (p2 , m(ζ)) is a monotonically increasing function of m(ζ) but, while it is concave-down
for m(ζ) . 0.1 GeV, it inflects thereafter to become concave-up. In addition, at fixed
m(ζ), M (p2 ) is a monotonically decreasing function of p 2 . It follows that a linear extrapolation determined by data on am ∈ [0.018, 0.072] will necessarily overestimate M (p 2 ) at all
positive values of p2 . The figures illustrate that the error owing to extrapolation increases
with increasing p2 and hence a significant overestimate can be anticipated on the domain
in which the condensate was inferred from lattice data.
It is straightforward to understand the behaviour evident in Figs. 5, 6 qualitatively.
The existence of DCSB means that in the neighbourhood of the chiral limit a massscale other than the current-quark mass determines the magnitude of M (p 2 ). As the
current-quark mass increases from zero, its magnitude will come to affect that of the mass
function. The gap equation is a nonlinear integral equation and hence this evolution of
the mass-dependence of M (p2 ) will in general be nonlinear. Only at very large values of
the current-quark mass will this scale dominate the behaviour of the mass function, as
seen in studies of heavy-quark systems [44], and the evolution become linear.
37
0.500
0.475
0.450
0.425
0.400
0.375
0.350
0.325
0.300
0.000
0.100
0.050
0.075
m(ξ = 19 GeV) (GeV)
0.025
0.125
Figure 6: Solid curve, calculated M (p 2 = 0), in GeV, as a function of the current-quark
mass m(ζ). The circles mark the current-quark masses in Eq. (3.12). Dashed-line, linear
interpolation of our result for M (p 2 = 0) on this mass domain.
3.7
Pion properties
The renormalised homogeneous Bethe-Salpeter equation (BSE) for the isovector-
pseudoscalar channel; i.e., the pion, is
Γjπ (k; P ) tu
=
Z
q
Λ
rs
[χjπ (q; P )]sr Ktu
(q, k; P ) ,
(3.29)
where k is the relative momentum of the quark-antiquark pair, P is their total momentum;
and
χjπ (q; P ) = S(q+ )Γjπ (q; P )S(q− ) ,
with Γjπ (k; P ) the pion’s Bethe-Salpeter amplitude, which has the general form
j
j
Γπ (k; P ) = τ γ5 iEπ (k; P ) + γ · P Fπ (k; P )
+ γ · k k · P Gπ (k; P ) + σµν kµ Pν Hπ (k; P ) .
(3.30)
(3.31)
In Eq. (3.29), K(q, k; P ) is the fully-amputated quark-antiquark scattering kernel, and
the axial-vector Ward-Takahashi identity requires that this kernel and that of the gap
38
equation must be intimately related. The consequences of this are elucidated in Refs.
[40, 41] and in the present case they entail
rs
Ktu
(q, k; P ) = −G((k − q)2 )
a a λ
λ
free
,
× Dµν (q − k)
γµ
γν
2
2
ts
ru
(3.32)
which provides the renormalisation-group-improved ladder-truncation of the BSE. The
efficacy of combining the renormalisation-group-improved rainbow-DSE and ladder-BSE
truncations is exhibited in Ref. [4]. In particular, it guarantees that in the chiral limit
the pion is both a Goldstone mode and a bound state of a strongly dressed quark and
antiquark, and ensures consistency with chiral low energy theorems [45, 77].
All the elements involved in building the kernel of the pion’s BSE were determined in
the last section and hence one can solve for the pion’s mass and Bethe-Salpeter amplitude.
To complete this exercise practically we consider
2
[1 − `(P )]
h
Γj5 (k; P )
i
tu
=
Z
q
Λ
rs
[χj5 (q; P )]sr Ktu
(q, k; P ) .
(3.33)
This equation has a solution for arbitrary P 2 and solving it one obtains a trajectory `(P 2 )
whose first zero coincides with the bound state’s mass, at which point Γ j5 (k; P ) is the true
bound state amplitude. In general, solving for `(P 2 ) in the physical domain: P 2 < 0,
requires that the integrand be evaluated at complex values of its argument. However,
as m2π M 2 (0); i.e., the magnitude of the zero is much smaller than the characteristic
dynamically-generated scale in this problem, we avoid complex arguments by adopting the
simple expedient of calculating `(P 2 > 0) and extrapolating to locate its timelike zero.
The Bethe-Salpeter amplitude is identified with the `(P 2 = 0) solution. Naturally, this
expedient yields the exact solution in the chiral limit; and in cases where a comparison
with the exact solution has been made for realistic, nonzero light-quark masses, the error
is negligible [78], as we shall subsequently illustrate.
39
mπ
fπ
fπ0
Calc. (quenched)
0.1385
0.066
0.063
Experiment
0.1385
0.0924
Table 1: Pion-related observables calculated using our lattice-constrained effective interaction. m(ζ = 19 GeV) = 3.3 MeV was chosen to give m π = 0.1395 GeV. The index “0”
indicates a quantity obtained in the chiral limit.
Once its mass and bound state amplitude are known, it is straightforward to calculate
the pion’s leptonic decay constant [43]:
ij
fπ δ 2 Pµ = Z2 tr
Z
q
Λ
τ i γ5 γµ χjπ (q; P ) .
(3.34)
In this expression, the factor of Z2 is crucial: it ensures the result is gauge invariant,
and cutoff and renormalisation-point independent. (The Bethe-Salpeter amplitude is normalised canonically [79].)
Table 1 lists values of pion observables calculated using the effective interaction obtained by fitting the quenched-QCD lattice data. (The nonzero current-quark mass in
the table corresponds to a one-loop evolved value of m(1 GeV) = 5.0 MeV.) To aid with
the consideration of these results we note that unquenched chiral-limit DSE calculations
that accurately describe hadron observables give [30] f π0 = 0.090 GeV. We infer from these
results that the pion decay constant in quenched lattice-QCD is underestimated by 30%.
Quantitatively equivalent results were found in Ref. [34].
The rainbow-ladder truncation of the gap and Bethe-Salpeter equations is chiral symmetry preserving: without fine-tuning it properly represents the consequences of chiral
symmetry and its dynamical breaking. The truncation expresses the model-independent
mass formula for flavor-nonsinglet pseudoscalar mesons [43] a corollary of which, at small
current-quark masses, is the Gell–Mann–Oakes–Renner relation:
(fπ0 )2 m2π = −2 m(ζ) hq̄qi0ζ + O(m2 (ζ)) .
(3.35)
40
Inserting our calculated values on the l.h.s. and r.h.s. of Eq. (3.35), we have
(0.093 GeV)4 cf. (0.091 GeV)4 ;
(3.36)
viz, the same accuracy seen in exemplary coupled DSE-BSE calculations; e.g., Ref. [29].
This establishes the fidelity of our expedient for solving the BSE.
Chapter 4
Algebraic model for the dressed quark-gluon vertex
Dyson-Schwinger equation predictions for the behaviour of dressed-gluon [27, 85, 86]
and dressed-quark propagators [29] have been confirmed in recent numerical simulations
of lattice-regularised QCD [87, 88]. Indeed, detailed study provides an understanding
of the circumstances in which pointwise agreement is obtained [38, 39]. This level of
sophistication does not prevail with the dressed-quark-gluon vertex, however. Acquiring
that is a realisable contemporary goal, and it is to aspects of this task that we address
ourselves herein.
4.1
Dressed quark-gluon vertex and the gap equation
This three-point Schwinger function can be calculated in perturbation theory but, since
we are interested in the role it plays in connection with confinement, DCSB and bound
state properties, that is inadequate for our purposes: these phenomena are essentially
nonperturbative.
Instead we begin by observing that the dressed vertex can be written
iΓaµ (p, q) =
i a
λ Γµ (p, q) =: l a Γµ (p, q) ;
2
(4.1)
viz., the colour structure factorises, and that twelve Lorentz invariant functions are required to completely specify the remaining Dirac-matrix-valued function; i.e.,
Γµ (p, q) = γµ Γ̂1 (p, q) + γ · (p + q) (p + q)µ Γ̂2 (p, q)
−i(p + q)µ Γ̂3 (p, q) + [. . .],
41
(4.2)
42
where the ellipsis denotes contributions from additional Dirac matrix structures that play
no part herein. Since QCD is renormalisable, the bare amplitude associated with γ µ
is the only function in Eq. (4.2) that exhibits an ultraviolet divergence at one-loop in
perturbation theory.
The requirement that QCD’s action be invariant under local colour gauge transforma2
tions entails [89]
kµ iΓµ (p, q) =
F g (k 2 ) [1 − B(p, q)] S(p)−1 − S(q)−1 [1 − B(p, q)] ,
(4.3)
wherein F g (k 2 ), k = p − q, is the dressing function that appears in the renormalised
covariant-gauge ghost propagator:
D g (k 2 , ζ 2 ) = −
F g (k 2 , ζ 2 )
,
k2
(4.4)
and B(p, q) is the renormalised ghost-quark scattering kernel. At one-loop order on the
domain in which perturbation theory is a valid tool
F g (k 2 , ζ 2 ) =
α(k 2 )
α(ζ 2 )
γg /β1
,
(4.5)
with the anomalous dimensions γg = −(3/8) C2 (G) in Landau gauge, which we use
throughout, and β1 = −(11/6) C2 (G) + (1/3) Nf , where C2 (G) = Nc and Nf is the
number of active quark flavours. This result may be summarised as F g (k 2 ) ≈ 1 on the
perturbative domain, up to ln p2 /Λ2QCD -corrections. In a similar sense, B(p, q) ≈ 0 in
Landau gauge on this domain.
2
3
NB. Equation (4.3) is equally valid when expressed consistently in terms of bare Schwinger functions.
An even closer analogy is a kindred result for Z(p2 ) in Eq. (2.21); viz., in Landau gauge [1−Z(p2 , ζ 2 )] ≡
0 at one loop in perturbation theory and hence, on the perturbative domain, corrections to this result are
modulated by ln ln p2 /Λ2QCD . This very slow evolution is exhibited, e.g., in the numerical results of Ref.
[14].
3
43
Equation (4.3) is a Slavnov-Taylor identity, one of a countable infinity of such relations
in QCD, and it is plainly an extension of the Ward-Takahashi identity that applies to the
fermion-photon vertex. The Ward-Takahashi identity entails that in the vertex describing
the coupling of a photon to a dressed-quark:
Γ̂γ1 (p, p) = A(p2 , ζ 2 ) ,
d
A(p2 , ζ 2 ) ,
dp2
d
Γ̂γ3 (p, p) = 2 2 B(p2 , ζ 2 ) .
dp
Γ̂γ2 (p, p) = 2
(4.6)
(4.7)
(4.8)
Identifying and understanding this nontrivial structure of the dressed-quark-photon vertex
has been crucial in describing the electromagnetic properties of mesons [90, 92, 91, 93].
The similarity between the Slavnov-Taylor and Ward-Takahashi identities has immediate, important consequences. For example, if the result
0 < F g (k 2 ) [1 − B(p, q)] < ∞
(4.9)
also prevails on the nonperturbative domain then, because of the known behaviour of the
dressed-quark propagator, Eq. (4.9) is sufficient grounds for Eq. (4.3) to forecast that in
the renormalised dressed-quark-gluon vertex
1 < Γ̂1 (p, p) < ∞
(4.10)
at infrared spacelike momenta. This result, an echo of Eq. (4.6), signals that the complete
kernel in the DSE satisfied by Γaµ (p, q) is attractive on the nonperturbative domain. The
ability to use the gap equation to make robust statements about DCSB rests on the existence of a systematic nonperturbative and chiral symmetry preserving truncation scheme.
One such scheme was introduced in Ref. [40]. It may be described as a dressed-loop
expansion of the dressed-quark-gluon vertex wherever it appears in the half amputated
44
dressed-quark-antiquark (or -quark-quark) scattering matrix: S 2 K, a renormalisationgroup invariant, where K is the dressed-quark-antiquark (or -quark-quark) scattering kernel. Thereafter, all n-point functions involved in connecting two particular quark-gluon
vertices are fully dressed.
The effect of this truncation in the gap equation, Eq. (2.20), is realised through the
following representation of the dressed-quark-gluon vertex [41]
+
Z1 Γµ (k, p) = γµ + L−
2 (k, p) + L2 (k, p) + [. . .] ,
(4.11)
with
L−
2 (k, p)
=
1
2Nc
Z
Λ
`
g 2 Dρσ (p − `)
×γρ S(` + k − p)γµ S(`)γσ ,
Z Λ
Nc
+
g 2 Dσ0 σ (`) Dτ 0 τ (` + k − p)
L2 (k, p) =
2
(4.12)
`
× γτ 0 S(p − `) γσ0 Γ3g
στ µ (`, −k, k − p) ,
(4.13)
wherein Γ3g is the dressed-three-gluon vertex. It is apparent that the lowest order contribution to each term written explicitly is O(g 2 ). The ellipsis in Eq. (4.11) represents terms
whose leading contribution is O(g 4 ); e.g., crossed-box and two-rung dressed-gluon ladder
diagrams, and also terms of higher leading-order.
The L−
2 term in Eq. (4.11) only differs from a kindred term in QED by the colour
factor. However, that factor is significant because it flips the sign of the interaction in this
channel with respect to QED; i.e., since
1
1 b
a b a
l l l =
l
C2 (G) − C2 (R) lb =
2
2Nc
N2 − 1
cf.
la 1c la = − C2 (R) 1c = − c
1c ,
2Nc
(4.14)
then single gluon exchange between a quark and antiquark is repulsive in the colouroctet channel. Attraction in the octet channel is provided by the L +
2 term in Eq. (4.11),
45
which involves the three-gluon vertex. These observations emphasise that Eq. (4.3) cannot be satisfied if the contribution from the three gluon vertex is neglected because the
Slavnov-Taylor identity signals unambiguously that on the perturbative domain there is
net attraction in the octet channel.
It is apparent, too, that the term involving the three gluon vertex is numerically amplified by a factor of Nc2 cf. the L−
2 (Abelian-like) vertex correction. Hence, if the integrals
are of similar magnitude then the Nc2 -enhanced three-gluon term must dominate in the
octet channel. This expectation is borne out by the one-loop perturbative calculation of
the two integrals exhibited in Eqs. (4.12), (4.13) and, moreover, the sum of both terms is
precisely that combination necessary to satisfy the Slavnov-Taylor identity at this order
[94].
4.2
Vertex and interaction model
In illustrating features of the nonperturbative and symmetry preserving DSE trunca-
tion scheme introduced in Ref. [40] in connection, for example, with DCSB, confinement
and bound state structure, Ref. [41] employed a dressed-quark-gluon vertex obtained by
resumming a subclass of diagrams based on L −
2 alone; namely, the vertex obtained as a
solution of
Γ−
µ (k+ , k− )
=
Z1−1 γµ
+
1
2Nc
Z
`
Λ
g 2 Dρσ (p − `)
× γρ S(`+ )Γ−
µ (`+ , `− )S(`− )γσ
.
(4.15)
It was acknowledged that this subclass of diagrams is 1/N c -suppressed but, in the absence
of nonperturbative information about L +
2 in general, and the dressed-three-gluon vertex
in particular, this limitation was accepted.
Herein we explore a model that qualitatively ameliorates this defect while preserving
characteristics that make calculations tractable and results transparent; viz., in Eq. (4.11)
46
we write
+
C
L−
2 + L2 ≈ L2 ,
(4.16)
where
LC2 (k, p) := − C C2 (R)
Z
`
Λ
g 2 Dρσ (p − `)
× γρ S(` + k − p)γµ S(`)γσ ,
(4.17)
and work with the vertex obtained as the solution of
ΓCµ (k+ , k− )
=
Z1−1 γµ
×
− C C2 (R)
Z
`
Λ
g 2 Dρσ (p − `)
γρ S(`+ )ΓCµ (`+ , `− )S(`− )γσ
.
(4.18)
To explain this model we remark that the parameter C is a global coupling strength
modifier. (NB. The value C = −(1/8) reproduces the vertex resummed in Ref. [41].) It
is introduced so that our Ansatz may mimic the effects of attraction in the colour-octet
channel without specifying a detailed form for the three-gluon vertex. This expedient
will give a faithful model so long as the integrals over the momentum dependence of L −
2
and L+
2 that appear in our calculations are not too dissimilar. This is plausible because:
they are both one-loop integrals projected onto the same Dirac and Lorentz structure and
hence are pointwise similar at this order in perturbation theory; and their direct sum
must conspire to give the simple momentum dependence in Eq. (4.3). Moreover, as we
shall demonstrate, the model has material illustrative capacity and that alone is sufficient
justification for proceeding.
As we have already noted, the value C = −(1/8) corresponds to completely neglecting
the three-gluon vertex term. There is also another particular reference case; namely, C = 1.
In this case a dressed-quark propagator obtained as the solution of the rainbow truncation
47
of the gap equation:
SR (p)−1 = Z2 (iγ · p + mbm )
Z Λ
λa
λa
g 2 Dµν (p − q) γµ SR (q) γν ,
+
2
2
q
(4.19)
when inserted in Eq. (4.18), yields a vertex Γ CR
µ (k, p) that satisfies
−1
(k − p)µ iΓCR
− SR (p)−1 ;
µ (k, p) = SR (k)
(4.20)
viz., a Ward-Takahashi identity. This is not materially useful, however, because herein we
will seek and work with a dressed-quark propagator that is not a solution of Eq. (4.19) but
rather a solution of Eq. (2.20) with a fully dressed vertex, and that vertex, determined self
consistently, will not in general satisfy Eq. (4.20).
4.2.1
Interaction model
A simplification, important to our further analysis, is a confining model of the dressedgluon interaction in Eq. (4.18). We use [42]
2
Dµν (k) := g Dµν (k) =
δµν
kµ kν
− 2
k
(2π)4 G 2 δ 4 (k) .
(4.21)
The constant G sets the model’s mass-scale and henceforth we mainly set G = 1 so that all
mass-dimensioned quantities are measured in units of G. Furthermore, since the model is
ultraviolet-finite, we will usually remove the regularisation mass-scale to infinity and set
the renormalisation constants equal to one.
In these things we follow Ref. [41]. In addition, we reiterate that the model defined by
Eq. (4.21) is a precursor to an efficacious class of models that employ a renormalisationgroup-improved effective interaction and whose contemporary application is reviewed in
Refs. [2, 3, 4]. It has many features in common with that class and, in addition, its
distinctive momentum-dependence works to advantage in reducing integral equations to
48
algebraic equations that preserve the character of the original. There is naturally a drawback: the simple momentum dependence also leads to some model-dependent artefacts,
but they are easily identified and hence not cause for concern.
4.3
Algebraic vertex and gap equations
If Eq. (4.21) is used in Eq. (4.18) then that part of the vertex which acts in the gap
equation has no dependence on the total momentum of the quark -antiquark pair; i.e., only
ΓCµ (p) := ΓCµ (p, p) contributes. In this case just the terms written explicitly in Eq. (4.2)
are supported in our model for the dressed vertex, which can be expressed
ΓCµ (p) = γµ αC1 (p2 ) + γ · p pµ αC2 (p2 ) − ipµ αC3 (p2 ) .
(4.22)
This is not a severe handicap because these Dirac structures are precisely those which
dominate in Eq. (4.3) if B(p, q) is not materially enhanced nonperturbatively. The vertex
equation is
ΓCµ (p) = γµ − C γρ S(p) ΓCµ (p) S(p) γρ ,
(4.23)
where we have used the fact that C2 (R) = 4/3 when Nc = 3.
Our analysis now mirrors that of Ref. [41]. The solution of Eq. (4.23) is:
ΓCµ (p)
=
∞
X
ΓCµ,i (p)
(4.24)
i=0
=
∞
X
i=0
γµ αC1,i (p2 ) + γ · p pµ αC2,i (p2 ) − i pµ αC3,i (p2 ) ,
(4.25)
where ΓCµ,i (p) satisfies a recursion relation:
ΓCρ,i+1 (p) = −C γµ S(p) ΓCρ,i (p) S(p) γµ ,
(4.26)
with ΓCµ,i=0 = γµ , the bare vertex, so that
αC1,0 = 1 , αC2,0 = 0 = αC3,0 .
(4.27)
49
In concert with Eq. (4.25), Eq. (4.26) yields an algebraic matrix equation (s = p 2 )


C
 α1,i+1 (s)


αCi+1 (s) :=  αC2,i+1 (s)


αC3,i+1 (s)
where (∆(s) = sA2 (s) + B 2 (s) )



 = O C (s; A, B) αCi (s) ,


O C (s; A, B) =

0
0
 −∆

2C 
− 2  2A2 sA2 − B 2
2AB
∆ 

4AB
4sAB
2(B 2 − sA2 )
It follows from Eqs. (4.25) and (4.28) that
!
∞
X
i
αC =
OC
αC0 =
i=0
(4.28)




.


1
αC
1 − OC 0
(4.29)
(4.30)
and hence, using Eq. (4.29),
αC1
αC2
αC3
=
∆
,
∆ − 2C
4CA2
(∆ − 4C)
,
∆2 + 2C(B 2 − sA2 ) − 8C 2 (∆ − 2C)
8CAB
= − 2
.
∆ + 2C(B 2 − sA2 ) − 8C 2
= −
(4.31)
(4.32)
(4.33)
With these equations one has a closed form for the dressed-quark-gluon vertex.
It is evident that the momentum dependence of the vertex is completely determined by
that of the dressed-quark propagator whose behaviour, however, the vertex itself influences
because it appears in the gap equation:
S(p)−1 = iγ · p + m + γµ S(p)ΓCµ (p) .
(4.34)
50
Subject to Eq. (4.22), the gap equation expresses two coupled algebraic equations:
A(s) = 1 +
sA2
1
[A (2αC1 − sαC2 ) − B αC3 ] ,
+ B2
(4.35)
B(s) = m +
sA2
1
[B (4αC1 + sαC2 ) − sA αC3 ] .
+ B2
(4.36)
The dressed-quark propagator and -quark-gluon vertex follow from the solution of these
equations, which is generally obtained numerically.
In the chiral limit, which here is simply implemented by setting m = 0, a realisation
of chiral symmetry in the Wigner-Weyl mode is always possible: it corresponds to the
B ≡ 0 solution of the gap equation. However, since the phenomena of QCD are built on
a Nambu-Goldstone realisation of chiral symmetry, we do not consider the Wigner-Weyl
mode any further. Its characterisation can be achieved in a straightforward manner by
adapting the analysis of Ref. [41] to our improved vertex model.
4.4
Algebraic results
At this point some observations are useful in order to establish a context for our
subsequent results. To begin we explore the ultraviolet behaviour of the model. It is
ultraviolet finite and hence at large spacelike momenta, s 1 (in units of G 2 )
a
A(s) ≈ 1 + ¯1 ,
s
b
B(s) ≈ m + ¯ 1 ,
s
(4.37)
with m the model’s finite current-quark mass. The model is useful because these results
persist in asymptotically free theories, up to ln p 2 /Λ2QCD -corrections. With this behaviour
51
it follows from Eqs. (4.31) – (4.37) that
2C
,
s
4C
αC2 (s) ≈ − 2 ,
s
8C
αC3 (s) ≈ − m 2 ,
s
αC1 (s) ≈ 1 +
(4.38)
(4.39)
(4.40)
and these results in turn mean that in the ultraviolet the behaviour of the massive dressedquark propagator is determined by the α 1 term in the vertex, so that
a1 = 2 , b 1 = 4 m .
¯
¯
(4.41)
The expansion of α1 (s) around 1/s = 0 reported in Eq. (4.38) is the same as that which
arises in QCD, apart from the usual ln(s/Λ 2QCD )- corrections. However, the leading terms
in α2,3 are different: on the perturbative domain in QCD these functions both begin with
a term of order (1/s)[ln(s/Λ2QCD )]d , with d some combination of anomalous dimensions.
The reason for the mismatch is readily understood. At one-loop order in QCD
1
ln(p2 /Λ2QCD )
B(p ) = m̂
2
2
γ1m /β1
,
(4.42)
with m̂ the renormalisation point invariant current-quark mass and γ 1m = (3/2) C2 (R).
(This makes explicit the logarithmic correction to the leading term in Eq. (4.37).) For
the purpose of this explanation then, on the perturbative domain, with F g (k 2 ) ≈ 1 and
B(p, q) ≈ 0, the Slavnov-Taylor identity, Eq. (4.3), is approximately equivalent to the
Ward-Takahashi identity. Hence, via Eq. (4.8),
α3 (s) ≈
m̂ γ1m
s β1
1
ln(p2 /Λ2QCD )
2
γ1m /β1 −1
.
(4.43)
It is thus evident that in QCD, even though they are not themselves divergent, the leading order terms in both α2,3 are induced by the momentum-dependent renormalisation of
52
elements contributing to their evaluation. Such terms are naturally missing in our ultraviolet finite model. The absence of 1/s terms in Eqs. (4.39) and (4.40) is therefore a model
artefact.
We observe in addition that with attraction in the colour-octet channel; namely, for
C > 0, (αC1 − 1) is necessarily positive on the perturbative domain and α C2,3 are negative.
These results are also true in QCD (α C1 > 1 up to logarithmic corrections). It would be an
exceptional result if these statements were not also true on the nonperturbative domain.
We now turn to the infrared domain and focus on s = 0 but consider 0 < m 1, in
which case
A(s = 0, m) ≈ a00 + a10 m , B(s = 0, m) ≈ b00 + b10 m .
(4.44)
Upon insertion of these expressions into the gap equation one obtains
√ 4 + 20C + 15C 2
2 + 3C
, a10 = − 2
,
2+C
(2 + C)5/2
√
1
2
b00 = 4 + 2 C , b10 = 0 2 =
;
2+C
(b0 )
a00 = 2
(4.45)
(4.46)
viz., results which show that in the neighbourhood of s = 0 and with attraction in the
colour-octet channel, A(s) decreases with increasing current-quark mass while B(s) increases. At this order the mass function is
M (s = 0, m) =
B(0, m)
= µ00 + µ10 m
A(0, m)
(4.47)
with
b0
= 00
a0
1
0
b
b a1
µ10 = 00 − 00 02
a0 (a0 )
µ00
=
=
√
3
2 (2 + C) 2
,
4 + 6C
6 + 23 C + 15 C 2
.
2 (2 + 3 C)2
For C > −(1/3) the mass function also increases with rising m.
(4.48)
(4.49)
53
The infrared behaviour of the dressed-quark-gluon vertex follows immediately via
Eqs. (4.31) – (4.33); using which one finds
αCi (s = 0, m) ≈ aCi,0 + aCi,1 m , i = 1, 2, 3,
(4.50)
with
aC1,0 = 1 +
C
,
2
1
C
,
aC1,1 = − √ √
8 2+C
C(2 − C)(2 + 3C)
,
aC2,0 = −
(2 + C)2
√
2 2 C 8 + 20 C + 2 C 2 − 9 C 3
C
,
a2,1 =
(2 + C)7/2
√
8C
C
a3,0 = − √
,
2+C
2 C (5 + 6 C)
aC3,0 =
.
(2 + C)2
(4.51)
(4.52)
(4.53)
(4.54)
(4.55)
(4.56)
These algebraic formulae provide a clear indication of the effect on the dressed-quarkgluon vertex of attraction in the projection of the quark-antiquark scattering kernel onto
the colour-octet channel. Attraction causes Γ̂1 (p, p) to be enhanced in the infrared cf. the
bare vertex; and it drives Γ̂2,3 (p, p) to magnified, negative values. These results also signal
that there are no surprises in the evolution into the infrared of the ultraviolet behaviour
expressed in Eqs. (4.38) – (4.40): in each case attraction ensures that a current-quark mass
acts to reduce the vertex function in magnitude.
4.5
Numerical results
4.5.1
Numerical results
In choosing a value for C we elect to be guided by results from quenched lattice-QCD
simulations of the dressed-quark propagator [88] and dressed-quark gluon vertex [37]. We
focus on a current-quark mass common to both simulations; namely, 60 MeV, at which
54
value the lattice dressed-quark propagator has [38]: Z qu (0) ≈ 0.7, Mqu (0) ≈ 0.42. Then, so
as to work with dimensionless quantities, we set m 60 = 0.06/Mqu (0) and, using Eqs. (4.44)
– (4.56), require a least-squares fit to
4
A(m60 ) = 1.4 ,
(4.57)
αC1 (0, m60 ) = 2.1 ,
(4.58)
−M (0, m60 )2 αC2 (0, m60 ) = 7.1 ,
(4.59)
−M (0, m60 ) αC3 (0, m60 ) = 1.0 .
(4.60)
C = C¯ = 0.51
(4.61)
This procedure yields
with an average relative error r̄ = 25% and standard deviation σ r = 70%. We note that
for C = 0.6: r̄ = 21%, σr = 72%, while for C = 0.4: r̄ = 31%, σr = 67%. If one omits
Eq. (4.59) from the fitting requirements then C¯ = 0.49 with r̄ = 2.5 % and σr = 63%. It
is evident that competing requirements bound the amount of attraction necessary in the
kernel. We can now illustrate the results for the dressed-quark propagator and dressedquark-gluon vertex.
A comparison of the curves in the upper panel of Fig. 7 shows clearly that the presence
of net attraction in the colour-octet quark-antiquark scattering kernel uniformly increases
the magnitude of A(p2 ) at all momenta. This effect is pronounced at infrared spacelike
momenta and particularly on the timelike domain, s < 0. In this and the following figures
C = 0 corresponds to the rainbow-ladder DSE truncation; i.e., the leading order term of
the truncation scheme introduced in Ref. [40].
In the lower panel one sees that on the spacelike domain, s > 0, the one-, two-,
three- and four-loop corrected vertices yield a result for A(p 2 ) that is little different from
4
There is a confusion of positive and negative signs in Ref. [37] concerning λ2 , λ3 , as defined therein.
Our signs are correct. With the conventions expressed in Eq. (4.22): 4λ2 = −α2 and 2λ3 = α3 .
55
5
A(s)
4
3
2
1
-4
-2
0
s
2
4
-2
0
s
2
4
5
A(s)
4
3
2
1
-4
Figure 7: Upper panel – C-dependence of A(s). For all curves m = 0.015. Solid line:
C = C¯ = 0.51; dash-dot-dot line: C = 1/4; dotted line: C = 0; and dash-dot curve:
¯ Solid line: complete
C = −1/8. Lower panel – Truncation-dependence of A(s), C = C.
solution; dash-dash-dot line - result obtained with only the i = 0, 1 terms retained in
Eq. (4.24), the one-loop corrected vertex; short-dash line - two-loop corrected; long-dash
line - three-loop corrected; and short-dash-dot line: four-loop corrected. In this and
subsequent figures, unless otherwise noted, dimensioned quantities are measured in units
of G in Eq. (4.21). A fit to meson observables requires G = 0.69 GeV and hence m = 0.015
corresponds to 10 MeV.
56
that produced by the completely resummed vertex. However, that is not true on the
timelike domain, whereupon confinement is expressed and hence nonperturbative effects
become important. In our model confinement is realised via the absence of a particlelike singularity in the dressed-quark propagator [11]. The cusp displayed by A(p 2 ) in the
timelike domain is one manifestation of this feature. The figure shows that convergence
to the solution obtained with the completely resummed vertex proceeds via two routes:
one followed by solutions obtained with an odd number of loop corrections to the vertex;
and another by those obtained using a vertex with an even number of corrections. This
effect is absent with net repulsion in the colour-octet projection of the quark-antiquark
scattering kernel.
We remark that A(p2 ) evolves slowly with the current-quark mass when that mass is
significantly smaller than the model’s mass-scale. However, when the current-quark mass
becomes commensurate with or exceeds that mass-scale, it acts to very effectively dampen
this function’s momentum dependence so that A(p 2 ) ≈ 1. This is also true in QCD.
We plot the dressed-quark mass function in Fig. 8. The existence of a nontrivial solution in the chiral limit is the realisation of DCSB, in our model and QCD. For current-quark
masses less than the model’s mass-scale, G, the dynamically generated mass determines
the scale of observables. However, for m & G, this explicit chiral symmetry breaking
mass-scale overwhelms that generated dynamically and enforces M (p 2 ) ≈ m. This is the
behaviour of the b-quark mass function in QCD [44].
From a comparison of the rainbow-ladder result, C = 0, in the lower panel with the
C = −1/8 and C¯ = 0.51 results, it is apparent that vertex dressing driven by net attraction
in the colour-octet scattering kernel reduces the magnitude of the mass function at infrared
momenta, a trend which is reversed for spacelike momenta s & G.
5
This effect has an
5
This pattern of behaviour is familiar from explorations [15, 17] of the effect in the gap equation of
vertex Ansätze [5, 95]; i.e., vertex models whose diagrammatic content is unknown but which exhibit
57
1.5
M(s)
1
0.5
0
-2
0
s
4
2
1.5
M(s)
1
0.5
0
-2
-1
0
s
1
2
Figure 8: Upper panel – Current-quark-mass-dependence of the dressed-quark mass function. For all curves C = C¯ = 0.51. Dotted line: m = m60 ; solid line: m = 0.015; dashed
line: chiral limit, m = 0. Lower panel – C-dependence of M (s). For all curves m = 0.015.
Solid line: C = C¯ = 0.51; dash-dot-dot line: C = 1/4; dotted line: C = 0; and dash-dot
curve: C = −1/8. In addition, for C = 0.51: dash-dash-dot line - M (s) obtained with
one-loop corrected vertex; and short-dash line - with two-loop-corrected vertex.
58
impact on the magnitude of the vacuum quark condensate. For example, the mapping of
Eq. (3.22) into our model is:
3
−hq̄qi = 2
4π
0
Z
s0
ds s
0
Z(s) M (s)
,
s + M (s)2
(4.62)
where s0 is the spacelike point at which the model’s mass function vanishes in the chiral
limit, and we find
−hq̄qi0C=C¯ = (0.231 G)3 = (0.16 GeV)3
(4.63)
with G = 0.69 GeV. The rainbow-ladder result is −hq̄qi 0C=0 = G 3 /(10π 2 ) = (0.15 GeV)3 so
that
hq̄qi0C=0
= 0.82 .
hq̄qi0C=C¯
(4.64)
This ratio drops to 0.50 when C = 1.0 is used to calculate the denominator.
It is thus evident that with attraction in the scattering kernel and at a common massscale, the condensate is significantly larger than that produced by a ladder vertex owing
to an expansion of the domain upon which the dressed-quark mass function has nonzero
support.
It is natural to ask for the pattern of behaviour in the presence of repulsion. In this
case Fig. 8 indicates that with C = −1/8 the value of the mass function is enhanced at
s = 0. The magnitude of the mass function grows larger still with a further decrease in C
and its domain of nonzero support expands. Therefore here, too, the condensate is larger
than with the ladder vertex; e.g.,
hq̄qi0C=0
= 0.92 ,
hq̄qi0C=− 1
8
and the ratio drops to 0.49 when C = −3/8 is used to evaluate the denominator.
properties in common with our calculated C > 0 result.
(4.65)
59
The implication of these results is that in general, with a given mass-scale and a common model dressed-gluon interaction, studies employing the rainbow-ladder truncation
will materially underestimate the magnitude of DCSB relative to those that employ a
well-constrained dressed-quark-gluon vertex. Naturally, in practical phenomenology, alterations of the mass-scale can compensate for this [15].
In Fig. 9 we portray the C-dependence of the scalar function associated with γ µ in the
dressed-quark-gluon vertex, αC1 (p2 ). It is particularly useful here to employ the rainbowladder result, C = 0, as our reference point because this makes the contrast between the
effect of attraction and repulsion in the colour-octet quark-antiquark scattering kernel
abundantly clear. Attraction uniformly increases the magnitude of α C1 (p2 ), while the opposite outcome is produced by omitting the effect of the three-gluon-vertex in the DSE for
the dressed-quark-gluon vertex. We remark that, as with A(p 2 ), αC1 (p2 ) evolves slowly with
the current-quark mass but again, when the current-quark mass becomes commensurate
with or exceeds the theory’s mass-scale, it acts to very effectively dampen the momentum
dependence of this function so that α C1 (p2 ) ≈ 1. This effect is apparent in the rainbow
vertex model employed in Ref. [38] to explain quenched-QCD lattice data.
Figure 10 illustrates the C-dependence of α C2 (p2 ), the scalar function modulating the
subleading Dirac vector component of the dressed-quark-gluon vertex. The qualitative
features of the completely resummed result for α C1 (p2 ) are also manifest here. However,
for this component of the vertex, which is purely dynamical in origin, there is a marked
difference at timelike momenta between the result obtained with an odd number of loop
corrections in the vertex and that obtained with an even number. We note that the
magnitude of this function also decreases with increasing current-quark mass.
It is notable that the size of our complete result for α C2 (p2 ) is an order of magnitude
smaller than that reported in Ref. [37]. This is an isolated case, however. The calculated
60
α1(s)
1.2
1
-4
-2
0
2
s
4
6
8
Figure 9: C-dependence of αC1 (s) in Eq. (4.22). For all curves m = 0.015. Solid line:
C = C¯ = 0.51; dash-dot-dot line: C = 1/4; dotted line: C = 0; and dash-dot curve:
C = −1/8. In addition, for C = 0.51: dash-dash-dot line - one-loop corrected α 1 (s); and
short-dash line - two-loop-corrected result.
0.5
α2(s)
0
-0.5
-1
-1.5
-4
-2
0
s
2
4
Figure 10: C-dependence of of αC2 (s). For all curves m = 0.015. Solid line: C = C¯ = 0.51;
dash-dot-dot line: C = 1/4; and dash-dot curve: C = −1/8. Moreover, for C = 0.51:
dash-dash-dot line - one-loop result for α C2 (s); short-dash line - two-loop result; long-dash
line - three-loop; and short-dash-dot line: four-loop. For C = 0, α C2 (s) ≡ 0.
61
magnitudes of the other functions in the dressed-quark propagator and -quark-gluon vertex
are commensurate with those obtained in quenched lattice-QCD. We remark in addition
that the lattice result is an order of magnitude larger than that obtained with a commonly
used vertex Ansatz [95]. This discrepancy deserves study in more sophisticated models.
In Fig. 11 we display what might be called the scalar part of the dressed-quark-gluon
vertex; viz., αC3 (p2 ). This is the piece of the vertex whose ultraviolet behaviour is most
sensitive to the current-quark mass. The figure demonstrates that at infrared momenta
αC3 (p2 ), too, is materially affected by the scale of DCSB, Eq. (4.63): at s = 0 the deviation
from its rainbow-truncation value is approximately four times that exhibited by α C1 (p2 ).
Hence, this term can be important at infrared and intermediate momenta.
Finally, since they are absent in rainbow truncation, it is illuminating to unfold the
different roles played by αC2 (s) and αC3 (s) in determining the behaviour of the gap equation’s
self-consistent solution. Some of these effects are elucidated in Fig. 12. The key observation
is that αC3 (s) alone is the source of all coupling between Eqs. (4.35) and (4.36) that is not
already present in rainbow-ladder truncation: α C3 B appears in the equation for A(s) and
αC3 s A appears in the equation for B(s). The action of α C2 is merely to modify the rainbowladder coupling strengths.
A consideration of Eqs. (4.35), (4.36) suggests that omitting α C3 (s) will affect A(s) at
infrared momenta but not B(s). That is easily substantiated by repeating the analysis
that gave Eqs. (4.45), (4.46), and is apparent in the figure. It will readily be appreciated
that neither αC2 (s) nor αC3 (s) can affect the deep ultraviolet behaviour of the gap equation’s
solution, Eqs. (4.37), (4.41), because they vanish too rapidly as 1/s → 0. This is plain in
Fig. 12.
At intermediate spacelike momenta both α C2 (s) and αC3 (s) are negative and hence act to
62
0
α3(s)
-0.5
-1
-1.5
-2
-1
0
s
1
2
3
0.5
0
α3(s)
-0.5
-1
-1.5
-2
-4
-2
0
s
2
4
Figure 11: Upper panel – Current-quark-mass-dependence of α C3 (s). For all curves C =
C¯ = 0.51. Dash-dot line: m = 2; dotted line: m = m 60 ; solid line: m = 0.015; dashed
line: chiral limit, m = 0. Lower panel – C-dependence of α C3 (s). For all curves m = 0.015.
Solid line: C = C¯ = 0.51; dash-dot-dot line: C = 1/4; and dash-dot curve: C = −1/8.
Moreover, for C = 0.51: dash-dash-dot line - one-loop result for α C3 (s); short-dash line two-loop result; long-dash line - three-loop; and short-dash-dot line: four-loop. For C = 0,
αC3 (s) ≡ 0.
63
5
A(s)
4
3
2
1
-4
-2
0
s
2
4
1
2
M(s)
1.5
1
0.5
0
-2
-1
0
s
¯ Eq. (4.61),
Figure 12: Upper panel – Impact of αC2 (s) and αC3 (s) on A(s). For C = C,
solid line: result obtained with both terms present; dashed-line: α C2 (s) omitted; dash-dot
line: αC3 (s) omitted. The dotted line is the result obtained with both terms present in the
vertex but C = −1/8. Lower panel – Impact of α C2 (s) and αC3 (s) on M (s). In all cases
m = 0.015.
64
magnify A(s) with respect to the value obtained using a bare vertex. However, they compete in Eq. (4.36): αC2 (s) works to diminish B(s) and αC3 (s) acts to amplify it. Therefore
in the absence of αC3 (s) one should expect M (s) = B(s)/A(s) to be suppressed at intermediate momenta, and consequently a condensate much reduced in magnitude. Omitting
αC2 (s) should yield the opposite effect. This is precisely the outcome of our numerical
studies:
hq̄qi0C=0 hq̄qi0C=C¯ = 1.97 ;
and
αC
3 ≡0
hq̄qi0C=0 hq̄qi0C=C̄ = 0.40 .
αC
2 ≡0
(4.66)
These aspects of our model provide an algebraic illustration of results obtained with
more sophisticated Ansätze, as apparent from a comparison with, for example, Refs. [15,
17].
4.6
Bethe-Salpeter Equation
4.7
Bethe-Salpeter Equation
The renormalised homogeneous Bethe-Salpeter equation (BSE) for the quark-
antiquark channel denoted by M can be compactly expressed as
[ΓM (k; P )]EF =
Z
Λ
q
[K(k, q; P )]GH
EF [χM (q; P )]GH
(4.67)
where: k is the relative momentum of the quark-antiquark pair and P is their total
momentum; E, . . . , H represent colour, flavour and spinor indices; and
χM (k; P ) = S(k+ ) ΓM (k; P ) S(k− ) ,
with ΓM (q; P ) the meson’s Bethe-Salpeter amplitude.
amputated dressed-quark-antiquark scattering kernel.
(4.68)
In Eq. (4.67), K is the fully-
65
4.7.1
Vertex consistent kernel
The preservation of Ward-Takahashi identities in those channels related to hadron
observables requires a conspiracy between the dressed-quark-gluon vertex and the BetheSalpeter kernel [40, 96]. The manner in which these constraints are realised for vertices
of the class considered herein was made explicit in Ref. [41]. In that systematic and
nonperturbative truncation scheme the rainbow gap equation and ladder Bethe-Salpeter
equation represent the lowest-order Ward-Takahashi identity preserving pair. Beyond this,
each additional term in the vertex generates a unique collection of terms in K, a subset
of which are always nonplanar.
For any dressed-quark-gluon vertex in the gap equation, which can be represented expressly by an enumerable series of contributions, the Bethe-Salpeter kernel that guarantees
the validity of all Ward-Takahashi identities is realised in
Z
Λ
ΓM (k; P ) =
Dµν (k − q) la γµ
q
a
a
× χM (q; P ) l Γν (q− , k− ) + S(q+ ) ΛM ν (q, k; P ) ,
(4.69)
where
ΛaM ν (q, k; P ) =
∞
X
n=0
Λa;n
M ν (q, k; P ) ,
(4.70)
66
with herein
1 a;n
Λ (`, k; P ) =
−
8 C Mν
Z
q
Λ
Dρσ (` − q) lb γρ χM (q; P )
× la ΓCν,n−1 (q− , q− + k − `) S(q− + k − `) lb γσ
Z Λ
+
Dρσ (k − q) lb γρ S(q+ + ` − k)
q
× la ΓCν,n−1 (q+ + ` − k, q+ ) χM (q; P ) lb γσ
Z Λ
0
Dρσ (` − q 0 )lb γρ S(q+
)
+
q0
0
× Λa;n−1
(q 0 , q 0 + k − `; P ) S(q−
+ k − `) lb γσ .
ν
(4.71)
This last equation is a recursion relation, which is to be solved subject to the initial
condition Λa;0
M ν ≡ 0.
The Bethe-Salpeter amplitude for any meson can be written in the form
ΓM (k; P ) = I c
NM
X
i=1
i
G i (k; P ) fM
(k 2 , k · P ; P 2 ) =: [G] f M ,
(4.72)
where G i (k; P ) are those independent Dirac matrices required to span the space containing
the meson under consideration. It then follows upon substitution of this formula that
Eq. (4.71) can be written compactly as
Λa;n
Mν =
n
io
h
a;n−1
C
i
+
L
Λ
KM
α
fM .
ν i;n−1
Mν
(4.73)
This states that Λa;n
M ν can be considered as a matrix operating in the space spanned by
the independent components of the Bethe-Salpeter amplitude, with its Dirac and Lorentz
structure projected via the contractions in the BSE. The first term (K M ) in Eq. (4.73)
represents the contribution from the first two integrals on the right-hand-side (r.h.s.)
of Eq. (4.71). This is the driving term in the recursion relation. The second term (L)
represents the last integral, which enacts the recursion.
67
4.7.2
Solutions of the vertex-consistent meson Bethe-Salpeter equation
π-meson
With the model of the dressed-gluon interaction in Eq. (4.21) the relative momentum
between a meson’s constituents must vanish. It follows that the general form of the BetheSalpeter amplitude for a pseudoscalar meson of equal-mass constituents is ( P̂ 2 = 1)
h
i
Γπ (P ) = γ5 if1π (P 2 ) + γ · P̂ f2π (P 2 ) .
(4.74)
To obtain the vertex-consistent BSE one must first determine Λ a;1
πν . That is obtained by
substituting Eq. (4.74) into the r.h.s. of Eq. (4.71). Only the first two integrals contribute
because of the initial condition and they are actually algebraic expressions when Eq. (4.21)
i in Eq. (4.73). Explicit calculation shows this to be identically zero,
is used. This gives Kπν
and hence Λa;1
πν ≡ 0. Since this is the driving term in the recursion relation then
Λaπν (k, k; P ) ≡ 0 .
(4.75)
While this result is not accidental [41], it is not a general feature of the vertex-consistent
Bethe-Salpeter kernel.
One thus arrives at a particularly simple vertex-consistent BSE for the pion (Q=P/2):
Γπ (P ) = − γµ S(Q) Γπ (P ) S(−Q) ΓCµ (−Q) .
(4.76)
Consider the matrices
i
4
P1 = − γ5 , P 2 =
1
4
γ · P̂ γ5 ,
(4.77)
which satisfy
fiπ (P ) = trD Pi Γπ (P ) .
(4.78)
They may be used to rewrite Eq. (4.76) in the form
f π (P ) = Hπ (P 2 )f π (P ) ,
(4.79)
68
Table 2: Calculated π and ρ meson masses, in GeV. (G = 0.69 GeV, in which case
m = 0.015 G = 10 MeV. In the notation of Ref. [40], this value of G corresponds to
η = 1.39 GeV.) n is the number of loops retained in dressing the quark-gluon vertex, see
Eq. (4.24), and hence the order of the vertex-consistent Bethe-Salpeter
√ kernel. NB. n = 0
corresponds to the rainbow-ladder truncation, in which case m ρ = 2 G, and that is why
this column’s results are independent of C.
n=0
MH
n=1
MH
n=2
MH
n=∞
MH
C = −(1/8)
π, m = 0
π, m = 0.01
ρ, m = 0
ρ, m = 0.01
0
0.149
0.982
0.997
0
0.153
1.074
1.088
0
0.154
1.089
1.103
0
0.154
1.091
1.105
C = (1/4)
π, m = 0
π, m = 0.01
ρ, m = 0
ρ, m = 0.01
0
0.149
0.982
0.997
0
0.140
0.789
0.806
0
0.142
0.855
0.871
0
0.142
0.842
0.858
C = C¯ = 0.51
π, m = 0
π, m = 0.01
ρ, m = 0
ρ, m = 0.01
0
0.149
0.982
0.997
0
0.132
...
...
0
0.140
0.828
0.844
0
0.138
0.754
0.770
wherein Hπ (P 2 ) is a 2 × 2 matrix
Hπ (P 2 )ij =
−
δ
trD Pi γµ S(Q) Γπ (P ) S(−Q) ΓCµ (−Q) .
δfjπ
(4.80)
Equation (4.79) is a matrix eigenvalue problem in which the kernel H is a function of
P 2 . This equation has a nontrivial solution if, and only if, at some M 2
det Hπ (P 2 ) − I P 2 +M 2 =0 = 0 .
(4.81)
The value of M for which this characteristic equation is satisfied is the bound state’s mass.
In the absence of a solution there is no bound state in this channel.
We have solved Eq. (4.81) for the pion and the results are presented in Table 2. That
the vertex-consistent Bethe-Salpeter kernel ensures the preservation of the axial-vector
Ward-Takahashi identity, and hence guarantees the pion is a Goldstone boson in the chiral
69
limit, is abundantly clear: irrespective of the value of C and the order of the truncation,
mπ = 0 for m = 0. Away from this symmetry-constrained point the results indicate that,
with net attraction in the colour-octet quark-antiquark scattering kernel, the rainbowladder truncation overestimates the mass; i.e., it yields a value greater than that obtained
with the fully resummed vertex (n = ∞). Moreover, the approach to the exact result for
the mass is not monotonic. On the other hand, given two truncations for which solutions
exist, characterised by n1 - and n2 -loop insertions, respectively, then
n2
n1
n=∞
n=∞
− MH
| , n 2 > n1 ;
| < |MH
|MH
− MH
(4.82)
viz., correcting the vertex improves the accuracy of the mass estimate.
ρ-meson
In our algebraic model the complete form of the Bethe-Salpeter amplitude for a vector
meson is
Γλρ (P ) = γ · λ (P ) f1ρ (P 2 ) + σµν λµ (P ) P̂ν f2ρ (P 2 ) .
(4.83)
This expression, which has only two independent functions, is simpler than that allowed by
a more sophisticated interaction, wherein there are eight terms. Nevertheless, Eq. (4.83)
retains the amplitudes that are found to be dominant in more sophisticated studies [30].
In Eq. (4.83), {λµ (P ); λ = −1, 0, +1} is the polarisation four-vector:
0
0
P · λ (P ) = 0 , ∀λ ; λ (P ) · λ (P ) = δ λλ .
(4.84)
The construction of the vertex-consistent ρ-meson BSE for the class of vertices under
consideration herein is fully described in Ref. [41]. The pion case illustrates the key modification. Brevity requires that we omit further details. Suffice to say, one arrives via a
mechanical procedure at the characteristic equation for the ρ-meson, which we solved.
70
The results are presented in Table 2. With increasing net attraction in the quarkantiquark scattering kernel the amount by which the rainbow-ladder truncation overestimates the exact mass also increases: with the amount of attraction suggested by lattice
data the n = 0 mass is 27% too large. A related observation is that the bound state’s
mass decreases as the amount of attraction between its constituents increases. Furthermore, with increasing attraction, even though the fully resummed vertex and consistent
kernel always yield a solution, there is no guarantee that a given truncated system supports
a bound state: the one-loop corrected vertex and consistent kernel (n = 1) do not have
sufficient binding to support a ρ-meson. This is overcome at the next order of truncation,
which yields a mass 9.7% too large. The observation that a given beyond-rainbow-ladder
truncation may not support a bound state, even though one is present in the solution
of the complete and consistent system, provides a valuable and salutary tip for model
building and hadron phenomenology.
Finally, as has often been observed, and independent of the truncation, bound state
solutions of gap-equation-consistent BSEs always yield the full amount of π-ρ mass splitting, even in the chiral limit. This splitting is driven by the DCSB mechanism. Its true
understanding therefore requires a veracious realisation of that phenomenon.
Dependence on the current-quark mass
In connection with this last observation it is relevant to explore the evolution with
current-quark mass of the pseudoscalar and vector meson masses, and of the difference
between them. The results for pseudoscalar mesons should be interpreted with the following caveat in mind. In constructing the vertex and kernel we omitted contributions from
gluon vacuum polarisation diagrams. These contribute only to flavour diagonal meson
channels. Hence, for light-quarks in the pseudoscalar channel, wherewith such effects may
71
Table 3: Current-quark masses required to reproduce the experimental masses of the
vector mesons. The values of mηc , mηb are predictions. Experimentally [100], m ηc =
2.9797 ± 0.00015 and mηb = 9.30 ± 0.03. NB. 0−
ss̄ is a fictitious pseudoscalar meson
composed of unlike-flavour quarks with mass m s , which is included for comparison with
other nonperturbative studies. All masses are listed in GeV.
mu,d = 0.01
mρ = 0.77
mπ = 0.14
ms = 0.166
mφ = 1.02
m0− = 0.63
ss̄
mc = 1.33
mJ/ψ = 3.10
mηc = 2.97
mb = 4.62
mΥ(1S) = 9.46
mηb = 9.42
be important [99, 98], our results should be understood to apply only to flavour nonsinglets. In principle, the same is true for light vector mesons. However, experimentally, the
ω and φ mesons are almost ideally mixed; i.e., the ω exhibits no s̄s content whereas the
φ is composed almost entirely of this combination. We therefore assume that the vacuum polarisation diagrams we have omitted are immaterial in the study of vector mesons.
(NB. It is an artefact of Eq. (4.21) that this model supports neither scalar nor axial-vector
meson bound states [42, 41].)
We fix the model’s current-quark masses via a fit to vector meson masses and the
results are presented in Table 3. The model we’re employing is ultraviolet finite and
hence our current-quark masses cannot be directly compared with QCD’s current-quark
mass-scales. Nevertheless, the values are quantitatively consistent with the pattern of
flavour-dependence in the explicit chiral symmetry breaking masses of QCD.
Our calculated results for the current-quark mass-dependence of pseudoscalar and
vector meson masses are presented in Fig. 13. In the neighbourhood of the chiral limit the
vector meson mass is approximately independent of the current-quark mass whereas the
pseudoscalar meson mass increases rapidly, according to (in GeV)
m20− ≈ 1.33 m m G ,
(4.85)
thereby reproducing the pattern predicted by QCD [43].
With the model’s value of the vacuum quark condensate, Eq. (4.63), this result allows
72
3
meson mass (GeV)
2.5
2
1.5
1
0.5
0
0
0.3
0.6
0.9
current-quark-mass (GeV)
1.2
8
meson mass (GeV)
4
2
1
0.5
0.25
0.125
0.01
0.1
current-quark-mass (GeV)
1
Figure 13: Evolution of pseudoscalar and vector q q̄ meson masses with the current-quark
mass. Solid line: pseudoscalar meson trajectory obtained with C = C¯ = 0.51, Eq. (4.61),
using the completely resummed dressed-quark-gluon vertex in the gap equation and the
vertex- consistent Bethe-Salpeter kernel; short-dash line: this trajectory calculated in
rainbow-ladder truncation. Long-dash line: vector meson trajectory obtained with C¯
using the completely resummed vertex and the consistent Bethe-Salpeter kernel; dash-dot
line: rainbow-ladder truncation result for this trajectory. The dotted vertical lines mark
the current-quark masses in Table 3.
73
one to infer the chiral-limit value of f π0 = 0.079 GeV via the Gell-Mann–Oakes–Renner
relation. It is a model artefact that the relative-momentum-dependence of Bethe-Salpeter
amplitudes is described by δ 4 (p) and so a direct calculation of this quantity is not realistic.
The value is low, as that of the condensate is low, because the model is ultraviolet finite.
In QCD the condensate and decay constant are influenced by the high-momentum tails of
the dressed-quark propagator and Bethe-Salpeter amplitudes [29, 30].
The curvature in the pseudoscalar trajectory persists over a significant domain of
current-quark mass. For example, consider two pseudoscalar mesons, one composed of
unlike-flavour quarks each with mass 2m s and another composed of such quarks with
mass ms . In this case
m20−
2ms
m20−
= 2.4 ,
(4.86)
ms
which indicates that the nonlinear evolution exhibited in Eq. (4.85) is still evident for
current-quark masses as large as twice that of the s-quark. With this result we reproduce
a feature of more sophisticated DSE studies [101, 102, 103] and a numerical simulation of
quenched lattice-QCD [104].
The mode of behaviour just described is overwhelmed when the current-quark mass
becomes large: m G. In this limit the vector and pseudoscalar mesons become degenerate, with the mass of the ground state pseudoscalar meson rising monotonically to meet
that of the vector meson. In our model
m1− = 1.04 ,
m0− m=mc
(4.87)
with a splitting of 130 MeV, and this splitting drops to just 40 MeV at m b ; viz., only 5%
of its value in the chiral limit. In addition to the calculated value, the general pattern
of our results argues for the mass of the pseudoscalar partner of the Υ(1S) to lie above
9.4 GeV. Indeed, we expect the mass splitting to be less than m J/ψ − mηc , not more. (See
74
1
0.6
2
2
mV - mPS (GeV)
2
0.8
0.4
0.2
0
0.01
0.1
current-quark-mass (GeV)
1
Figure 14: Evolution with current-quark mass of the difference between the squaredmasses of vector and pseudoscalar mesons ( C¯ = 0.51) using the completely resummed
dressed-quark-gluon vertex in the gap equation and the vertex-consistent Bethe-Salpeter
kernel. The dotted vertical lines mark the current-quark masses in Table 3.
also; e.g., Ref. [105].)
In Fig. 14 one observes that on a material domain of current-quark masses: m 21− −
m20− ≈ 0.56 GeV2 , an outcome consistent with experiment that is not reproduced in
numerical simulations of quenched lattice-QCD [104]. The difference is maximal in the
vicinity of mc , a result which re-emphasises that heavy-quark effective theory is not an
appropriate tool for the study of c-quarks [44].
Figure 15 is instructive. It shows that with growing current-quark mass the rainbowladder truncation provides an increasingly accurate estimate of the ground state vector
meson mass. At the s-quark mass the relative error is 20% but that has fallen to < 4% at
the c-quark mass.
Similar statements are true in the valid pseudoscalar channels. In fact, in this case
75
0.3
(ladder - full) / full
0.25
0.2
0.15
0.1
0.05
0
0.0001
0.001
0.01
0.1
current-quark-mass (GeV)
1
Figure 15: Evolution with current-quark mass of the relative difference between the meson
mass calculated in the rainbow-ladder truncation and the exact value. Solid lines: vector
meson trajectories; and dashed-lines; pseudoscalar meson trajectories. The dotted vertical
lines mark the current-quark masses in Table 3. We used C¯ = 0.51.
the agreement between the truncated and exact results is always better; e.g., the absolute
difference reaches its peak of ≈ 60 MeV at m ∼ 4 m s whereat the relative error is only 3%.
This behaviour is fundamentally because of Goldstone’s theorem, which requires that all
legitimate truncations preserve the axial-vector Ward-Takahashi identity and hence give
a massless pseudoscalar meson in the chiral limit. It is practically useful, too, because
it indicates that the parameters of a model meant to be employed in a rainbow-ladder
truncation study of hadron observables may reliably be fixed by fitting to the values of
quantities calculated in the neighbourhood of the chiral limit.
The general observation suggested by Fig. 15 is that with increasing current-quark mass
the contributions from nonplanar diagrams and vertex corrections are suppressed in both
the gap and Bethe-Salpeter equations. Naturally, they must still be included in precision
76
spectroscopic calculations. It will be interesting to reanalyse this evolution in a generalisation of our study to mesons composed of constituents with different current-quark masses,
and thereby extend and complement the limited such trajectories in Refs. [101, 102].
Chapter 5
Quark-gluon vertex model and lattice-QCD data
In the absence of well-constrained nonperturbative models for the vertex, it has often
been assumed that a reasonable beginning is the Ball-Chiu [95] or Curtis-Pennington [5]
Abelian Ansatz times the appropriate color matrix. An example is provided by the recent results from a truncation of the gluon-ghost-quark DSEs where this vertex dressing
contributes materially to a reasonable quark condensate value [18]. However, there is
no known way to develop a Bethe-Salpeter (BSE) kernel that is dynamically matched
to a quark self-energy defined in terms of such a phenomenological dressed vertex in the
sense that chiral symmetry is preserved through the axial-vector Ward-Takahashi identity. The latter implementation of chiral symmetry guarantees the Goldstone boson nature of the flavor non-singlet pseudoscalars independently of model details [43]. There is
a known constructive scheme [40] that defines a diagrammatic expansion of the BSE kernel corresponding to any diagrammatic expansion of the quark self-energy such that the
axial-vector Ward-Takahashi identity is preserved. For this reason, recent nonperturbative
vertex models have employed simple diagrammatic representations [19, 20, 110, 111].
It is only recently that lattice-QCD has begun to provide information on the infrared
structure of the dressed quark-gluon vertex [37, 112]. In this work we generate a model
dressed vertex, for zero gluon momentum, based on an Ansatz for non-perturbative extensions of the only two diagrams that contribute at 1-loop order in perturbation theory.
An existing ladder-rainbow model kernel is the only required input. We compare to the
recent lattice-QCD data without parameter adjustment.
77
78
5.1
One-loop perturbative vertex
In Euclidean metric we denote the dressed-quark-gluon vertex for gluon momentum
k and quark momentum p by ig tc Γσ (p + k, p), where tc = λc /2 and λc is an SU(3) color
matrix. Through O(g 2 ), i.e., to 1-loop, the amplitude Γσ is given, in terms of Fig. 16, by
NA
Γσ (p + k, p) = Z1F γσ + ΓA
σ (p + k, p) + Γσ (p + k, p) + . . . ,
i
(5.1)
with
ΓA
σ (p
+ k, p) = −(CF −
CA
)
2
Z
Λ
q
g 2 Dµν (p − q)γµ
×S0 (q + k) γσ S0 (q)γν
,
(5.2)
and
ΓNA
σ (p
+ k, p) =
C
− A
2
Z
Λ
g 2 γµ S0 (p − q)γν Dµµ0 (q + k)
q
3g
×iΓµ0 ν 0 σ (q
+ k, q) Dν 0 ν (q)
,
(5.3)
Here Z1F (µ2 , Λ2 ) is the vertex renormalization constant to ensure Γ σ = γσ at renormalization scale µ. The following quantities are bare: the three-gluon vertex ig f abc Γ3g
µνσ (q+k, q),
the quark propagator S0 (p), and the gluon propagator Dµν (q) = Tµν (q)D0 (q 2 ), where
Tµν (q) is the transverse projector. The next order terms in Eq. (5.1) are O(g 3 ): the contribution involving the four-gluon vertex, and O(g 4 ): contributions from crossed-box and
two-rung gluon ladder diagrams, and 1-loop dressing of the triple-gluon vertex, etc.
The color factors in Eqs. (5.2) and (5.3), given by
ta tb ta = (CF −
ta f abc tb =
CA c
it
2
CA b
)t
2
=
=−
Nc c
it
2
1 b
t
2Nc
,
(5.4)
reveal two important considerations. The color factor of the (Abelian-like) term Γ A
σ would
be given by ta ta = CF = (Nc2 − 1)/2Nc for the strong dressing of the photon-quark vertex,
79
i.e., in the color singlet channel. The octet Γ A
σ is of opposite sign and is suppressed by
a factor 1/(Nc2 − 1): single gluon exchange between a quark and antiquark has relatively
weak repulsion in the color-octet channel, compared to strong attraction in the colorsinglet channel. Net attraction for the gluon vertex (at least to this order) is provided
by the non-Abelian ΓNA
term, which involves the three-gluon vertex: the color factor is
σ
amplified by −Nc2 over the ΓA
σ term.
A
NA
Figure 16: The quark-gluon vertex at one loop. The left diagram labelled A is the AbelianNA
like term ΓA
σ , and the right diagram labelled NA is the non-Abelian term Γ σ .
The specific form of the bare triple-gluon vertex is conveniently expressed in terms
of three momenta p1 = q + k, p2 = −q and p3 = −k, that are outgoing. Thus with
3g
Γ3g
µνσ (q + k, q) ≡ Γ̃µνσ (p1 , p2 , p3 ), we have
Γ̃3g
µνσ (p1 , p2 , p3 ) = − (p1 − p2 )σ δµν + (p2 − p3 )µ δνσ
+ (p3 − p1 )ν δσµ
,
(5.5)
and the complete vertex is symmetric under permutations of all gluon coordinates. In
Landau gauge Γ3g
µνσ obeys the Slavnov-Taylor identity
−1
−1
kσ Γ3g
µνσ (q + k, q) = D0 (q) Tµν (q) − D0 (q + k) Tµν (q + k) .
(5.6)
Our nonperturbative model addresses the k = 0 case and makes an extension of the bare
result
Γ3g
µνσ (q, q) = −
∂
D −1 (q) Tµν (q)
∂qσ 0
,
(5.7)
80
which allows the amplitude for the non-Abelian diagram at k = 0 to take the form
ΓNA
σ (p, p)
CA
2
Z
Λ
= −i
γµ S0 (p − q)γν
q
o
n
∂ 2
g D0 (q 2 ) Tµν (q)
×
.
∂qσ
(5.8)
It is easy to verify that the Abelian diagram gives
ΓA
σ (p, p) = −i[1 −
CA −1 ∂
CF ]
Σ(1) (p)
2
∂pσ
,
(5.9)
in terms of the 1-loop self-energy.
NA yields a vertex that satisfies the
The dressing provided by the combination Γ A
σ + Γσ
Slavnov-Taylor identity (STI) through O(g 2 ) [94]. This identity expresses the divergence
of the vertex in terms of the bare and 1-loop contributions to three objects: S(p) −1 ,
the ghost propagator dressing function, and the ghost-quark scattering amplitude. The
1-loop S(p)−1 part of this relation is generated partly from Γ A
σ (with a weak repulsive
(with the complementary strongly attractive color
color strength) and partly from ΓNA
σ
2
strength). The ΓNA
σ term also provides the explicitly non-Abelian terms of the O(g ) STI.
5.2
Nonperturbative vertex model
Our nonperturbative model for the dressed quark-gluon vertex is defined by extentions
of Eqs. (5.2) and (5.3) into dressed versions determined solely from an existing ladderrainbow model DSE kernel. The bare quark propagators in Eqs. (5.2) and (5.3) are
replaced by solutions of the quark DSE in rainbow truncation, namely,
S(p)−1 = Z2 i p
/ + Z4 m(µ)
Z Λ
G(q 2 )
Tµν (q) γµ S(p0 ) γν ,
+ CF
2
q
0
p
(5.10)
81
λ1(p) Quenched Lattice
2.5
2
4p λ2(p) Quenched Lattice
-2pλ3(p) Quenched Lattice
λ1(p) DSE--Lat (quenched)
2
2
4p λ2(p) DSE--Lat
-2pλ3(p) DSE--Lat
1.5
λ1(p) Abelian Ansatz (WI)
-2pλ3(p) Abelian Ansatz (WI)
1
0.5
0
0
1
p (GeV)
2
3
Figure 17: The amplitudes of the dressed quark-gluon vertex at zero gluon momentum and
for quark current mass m(µ = 2 GeV) = 60 MeV. Quenched lattice data [37] is compared
to the results of the DSE-Lat model [38]. The Abelian Ansatz (Ward identity) is also
shown except for λ2 (p) which is almost identical to the DSE-Lat model.
where q = p − p0 . A particular ladder-rainbow kernel is specified by the effective quarkquark coupling G(q 2 ). Two different DSE models are employed and both have the ultraviolet behavior specified by QCD with 1-loop renormalization group improvement, i.e., the 1loop renormalizations of the quark and gluon propagators and the pair of quark-gluon vertices have been absorbed so that G(q 2 ) matches 4 π αs1−loop (q 2 ) = 4π 2 γm /ln(q 2 /Λ2QCD ) [29].
Here γm = 12/(33 − 2Nf ) is the anomalous mass dimension which arises in the leading
logarithmic behavior of the quark mass function in the ultraviolet. The two DSE models
differ in the infrared content of G(q 2 ) specified by parameterization.
The first model (DSE-Lat) [38] is defined by
G(q 2 )
= Dlat (q 2 )Γ1 (q 2 , m(µ))
q2
,
(5.11)
82
where Dlat (q 2 ) is a fit to quenched lattice data for the Landau gauge gluon propaga13
tor [21] that has the correct 1-loop logarithmic behavior ∼ ln(q 2 /Λ2QCD )− 22 in the ultraviolet. In the infrared Dlat (q 2 ) is finite and is suppressed with respect to the bare
propagator. The vertex factor γν Γ1 (q 2 , m(µ)) represents the remaining 1-loop renormalizations for ultraviolet matching to 4 π α s1−loop (q 2 )/q 2 (with Nf = 0) and also contains a
parameterized representation of the remaining infrared dressing. Explicit formulas are
given in Ref. [38]. Parameters are determined by requiring that the DSE solutions reproduce the quenched lattice data [88] for S(p) in the available domain p 2 < 10 GeV 2
and m(µ = 2 GeV) < 200 MeV. In this sense, the DSE-Lat model represents quenched
dynamics. It is found that the necessary vertex dressing is a strong but finite enhancement. The model easily reproduces m π with a current mass that is within acceptible
limits. However the resulting chiral condensate hq̄qi 0µ=1
GeV
= (0.19 GeV)3 is a factor of
2 smaller than the value (0.24 ± 0.01 GeV) 3 from a best fit [113] of strong interaction
observables [38]. This is attributed to the quenched approximation in the lattice data.
The second model (DSE-MT) [30] implements a one-parameter representation for the
infrared sector of G(q 2 ) that is fit to the empirical chiral condensate. The explicit form is
G(q 2 )
q2
=
+
4π 2 D q 2 −q2 /ω2
e
ω6
4π 2 γm F(q 2 )
2 ,
1
2
2
2 ln τ + 1 + q /ΛQCD
(5.12)
Here the first term implements the infrared enhancement necessary to generate the empir−s
ical condensate, while the second term, with F(s) = (1 − exp( 4m
2 ))/s, connects smoothly
t
with the 1-loop renormalization group behavior of QCD. Apart from the fixed values
mt = 0.5 GeV, τ = e2 − 1, Nf = 4, and ΛQCD = 0.234 GeV, the free parameters, ω and D
are not independent. The fitted observables are essentially constant along the trajectory
ωD = (0.72 GeV)3 for ω = 0.3 − 0.5 GeV. A standard choice is ω = 0.4 GeV and D = 0.93
83
GeV2 . The model provides an excellent description of a wide variety of light quark meson physics including the masses and decay constants of the light-quark pseudoscalar and
vector mesons [29, 30], the elastic charge form factors F π (Q2 ) and FK (Q2 ) [114], and the
electroweak transition form factors of the pseudoscalars and vectors [115, 116]. In this
sense it represents unquenched dynamics.
In the ultraviolet, the q̄q scattering kernel appearing in the Abelian-like vertex diagram
shown in Fig. 16-A coincides with the ladder-rainbow kernel; thus the latter provides a
suitable nonperturbative exension. We substitute g 2 D0 (q 2 ) → G(q 2 )/q 2 in the integrand
for ΓA
σ , in Eq. (5.2). The vertex for the external gluon is taken to be bare.
Even in the ultraviolet, the q̄q scattering kernel does not appear explicitly in the nonAbelian vertex diagram shown in Fig. 16-NA. However at k = 0, the expression in Eq. (5.8)
for ΓNA
σ (p, p) has combined the triple gluon vertex and the gluon propagators to produce a
form that emphasizes the close connection to the ladder kernel and the self-energy integral.
The same nonperturbative extension g 2 D0 (q 2 ) → G(q 2 )/q 2 now suggests itself for Fig. 16NA, and we use it. Justifications for this choice are consistency and simplicity; no new
parameters are introduced.
5.3
Results and discussion
The general nonperturbative vertex at k = 0 has a representation in terms of three
invariant amplitudes; here we choose
Γσ (p, p) = γσ λ1 (p2 ) − 4pσ γ · p λ2 (p2 ) − i2pσ λ3 (p2 )
.
(5.13)
since the lattice-QCD data [37] is provided in terms of these λ i (p2 ) amplitudes. A useful
comparison is the corresponding vertex in an Abelian theory like QED; it is given by the
I
−1 (p)/∂p in terms of the exact propagator S −1 (p). With
Ward identity ΓW
σ
σ (p, p) = −i∂S
WI
0
S −1 (p) = iγ · p A(p2 ) + B(p2 ), this leads to the correspondance λ WI
1 = A, λ2 = −A /2,
84
λ1(p) DSE--lat (quenched)
2.5
2
4p λ2(p) DSE-Lat (quenched)
-2pλ3(p) DSE-Lat (quenched)
2
λ1(p) DSE--MT (unquenched)
2
4p λ2(p) DSE--MT (unquenched)
-2pλ3(p) DS-MT (unquenched)
1.5
1
0.5
0
0
1
p (GeV)
2
3
Figure 18: The amplitudes of the dressed quark-gluon vertex at zero gluon momentum,
and for quark current mass m(µ = 2 GeV) = 60 MeV, from two models: DSE-Lat [38]
and DSE-MT [30] that relate to quenched and unquenched content respectively.
0
0
2
2
and λWI
3 = B , where f = ∂f (p )/∂p .
In Fig. 17 we display the DSE-Lat model results in a dimensionless form for comii
parison with the (quenched) lattice data . The renormalization scale of the lattice data
is µ = 2 GeV where λ1 (µ) = 1, A(µ) = 1. We compare to the lattice data set for which
m(µ) = 60 MeV. The same renormalization scale and conditions have been implemented
iii
for both DSE models . For λ1 and λ3 we also compare with the Abelian Ansatz in which
the amplitudes are obtained from the quark propagator through the Ward Identity, which
is equivalent to the k = 0 limit of either the Ball-Chiu [95] or Curtis-Pennington [5] Ansatz.
Without parameter adjustment, the model reproduces the lattice data for λ 1 and λ3 quite
ii
We note that in Ref. [37] both the lattice data, and the Abelian (Ward identity) Ansatz, for λ3 (p) are
presented as positive. These two sign errors have been acknowledged [112].
iii
To facilitate change of the scale µ, we have slightly modified both DSE kernels (both originally defined
at fixed scale µ0 = 19 GeV) by including the additional kernel strength factor Z22 (µ2 , Λ2 )/Z22 (µ20 , Λ2 )
recommended by Maris [106]. This does not alter results for observables.
85
well over the whole momentum range for which data is available. The Abelian Ansatz,
while clearly inadequate for λ1 below 1.5 GeV, reproduces λ3 . The present lattice data
for λ2 has large errors; it suggests infrared strength that is seriously underestimated by
the model. (The Abelian Ansatz for λ 2 is very close to the DSE model and for reasons of
clarity, is not displayed.)
The relative contributions to the vertex dressing made by Γ NA
and ΓA
σ
σ are indiA
NA
A
cated by the following amplitude ratios at p = 0: λ NA
1 /λ1 = −60, λ2 /λ2 = −14, and
A
NA
λNA
3 /λ3 = −12. Thus the non-Abelian term Γ σ dominates to a greater extent than what
the ratio of color factors (−9) would suggest; it also distributes its infrared strength to
NA
favor λ1 more so than does ΓA
σ . Since the momentum-dependent shapes of the λ i (p) and
λA
i (p) are quite similar, the present model results could be summarized quite effectively
NA up by about 10%.
by ignoring ΓA
σ and scaling Γσ
Due to the definition of the two DSE models, their comparison in Fig. 18 provides an
estimate of the effects of the quenched approximation. The effects are moderate within the
present DSE model framework. Fig. 18 also suggests that a model including the four gluon
vertex as well as the two diagrams of Fig. 16 should be considered, especially for amplitude
λ2 . The question of the importance of the iterations of the diagrams of Fig. 16 also arises.
We have estimated such effects by iteration to all orders based on the ladder-rainbow
kernel. This amounts to solution of a ladder Bethe-Salpeter integral equation in which
the inhomogeneity is our dressed extension of Z 1F γσ + ΓNA
σ (p, p) and the kernel term is
the dressed extension of ΓA
σ (p, p) with the internal γσ replaced by Γσ (q, q). This generates
very little change—significantly less than the quenching effect evident in Fig. 18. This is
due to the small color factor of the kernel term. We have not explored the consequences
of using the dressed vertex self-consistently for the internal quark-gluon vertices of Γ NA
σ
in Fig. 16-NA.
86
The nonperturbative Ansatz we have applied to Eq. (5.8) for the non-Abelian diagram,
Fig. 16-NA, effectively includes dressing of the triple-gluon vertex Γ 3g
µνσ . Some perturbative
studies of Γ3g
µνσ have been made at 1-loop [107, 108] but they provide no guidance for
extension to infrared scales. The nonperturbative Ansätze for Γ 3g
µνσ suggested in Refs. [3]
and [109] for use within truncated gluon-ghost-quark DSEs require explicit models for the
ghost dressing function and the ghost-gluon vertex that appear in the STI for Γ 3g
µνσ . Such
considerations are beyond the scope of the present work; they would entail additional
parameters that are not warranted at this stage.
A different approach to the nonperturbative extension of the non-Abelian diagram,
Fig. 16-NA, has recently been explored in Refs. [110] and [111]. That approach employs
a bare triple-gluon vertex and dressed gluon two-point functions resulting from previous
solution of a truncation of the coupled ghost-gluon-quark DSEs [18]. (The quark-gluon
vertex within that calculation was described by the Curtis-Pennington [5] Abelian Ansatz
times the square of the infrared enhanced ghost dressing function.) The strong infrared
suppression inherent in such gluon propagator solutions produces a very weak quark-gluon
vertex unless the internal quark-gluon vertices of Fig. 16-NA are also enhanced. Refs. [110]
and [111] proceed by assuming that attachment of a single ghost dressing function to each
vertex is appropriate for this. The results are similar to the present work, except that the
m(µ) = 115 MeV case is considered [110, 111].
The infrared content of QCD 2-point and 3-point functions is not a settled subject
and there is much to be gained from comparison of a variety of modeling strategies. Our
approach to the vertex differs from Refs. [110] and [111] in the following respect. We exploit
the similarity betwen Eq. (5.8) for Γ NA
σ (p, p) and a zero momentum gluon insertion into
the lowest order q̄q scattering kernel appearing in the rainbow diagram for the quark selfenergy. Note that if the derivative in Eq. (5.8) were to act also on the transverse projector,
87
then result would be an Abelian-like derivative of the self-energy. The correction terms
to this are evidently small for the resulting amplitudes λ 2 and λ3 but they are large for
λ1 . Since the renormalization group improved DSE-Lat kernel has infrared content that
supplements quenched lattice data for the gluon 2-point function to get a precise fit to
the quenched-lattice quark propagator mass-function, it is not surprizing that our result
for λ3 is a closer representation of the lattice data than the corresponding result from
Refs. [110] and [111]. The latter works do not determine parameters by fitting the quark
propagator.
Our approach induces effective dressing of the gluon propagators, internal quark vertices and the triple-gluon vertex in one quantity that is tightly constrained by quenched
lattice data for 2-point functions. There is no well-defined and consistent way to separate
the various contributions. If we assume this q̄q kernel can be used for each quark-gluon
interaction in Fig. 16-NA then our Ansatz is equivalent to corresponding use of a dressed
2
triple-gluon vertex Γ3g
µνσ satisfying Eqs. (5.6) and (5.7) with the substitution D 0 (q ) →
[G(q 2 )/g 2 (µ2 )]/q 2 . If one were to replace this by the bare vertex, leaving other factors unchanged, then the final vertex amplitudes increase by at least an order of magnitude; the
infrared dressing enhancements have been treated inconsistently. An opposite extreme,
that is at least consistent, is to use the bare limit for all elements of Fig. 16-NA except
for the quark propagator. That is, use D 0 (q 2 ) → 1/q 2 in Eq. (5.8). This underestimates
the quark-gluon vertex amplitudes by an order of magnitude. In the absence of clear
guidance for the infrared content of each of the 2-point and 3-point functions involved, the
consistent use of an empirical ladder-rainbow kernel representation of the q̄q scattering
amplitude recommends itself.
An issue of consistency that requires future attention is the following. Both lattice data
and the present model calculation indicate that vertex amplitude λ 1 is infrared enhanced to
88
about 2.5 times the bare value. However the phenomenological vertex enhancement found
in the DSE-Lat model (assumed to be concentrated solely in that single amplitude) is about
6 times this at a typical infrared point p 2 ∼ 0.04 GeV 2 [117]. The proper distribution of
vertex strength over the many amplitudes available at finite momenta may be part of
resolution. On a more general note, it would be desirable to replace the phenomenological
aspects of the dressing effects implicitly included for the internal quark-gluon vertices by
the results of explicit model calculations.
Chapter 6
Summary and Conclusions
We studied quenched-QCD using a rainbow-ladder truncation of the Dyson-Schwinger
equations (DSEs) and demonstrated that existing results from lattice simulations of
quenched-QCD for the dressed-gluon and -quark Schwinger functions can be correlated
via a gap equation that employs a renormalisation-group-improved model interaction. As
usual, the ultraviolet behaviour of this effective interaction is fully determined by perturbative QCD.
For the infrared behaviour we employed an Ansatz whose parameters were fixed in a
least squares fit of the gap equation’s solutions to lattice data on the dressed-quark mass
function, M (p2 ), at available current-quark masses. With our best-fit parameters the mass
functions calculated from the gap equation were indistinguishable from the lattice results.
The gap equation simultaneously yields the dressed-quark renormalisation function, Z(p 2 ),
and, without tuning, our results agreed with those obtained in the lattice simulations.
To correlate the lattice’s dressed-gluon and -quark Schwinger functions it was necessary
for the gap equation’s kernel to exhibit infrared enhancement over and above that observed
in the gluon function alone. We attributed that to an infrared enhancement of the dressedquark-gluon vertex. The magnitude of the vertex modification necessary to achieve the
correlation is semi-quantitatively consistent with that observed in quenched lattice-QCD
estimates of this three-point function.
With a well-defined effective interaction, the gap equation provides a solution for
the dressed-quark Schwinger function at arbitrarily small current-quark masses and, in
89
90
particular, in the chiral limit: no extrapolation is involved. A kernel that accurately
describes dressed-quark lattice data at small current-quark masses may therefore be used
as a tool with which to estimate the chiral limit behaviour of the lattice Schwinger function.
Our view is that this method is a more reliable predictor than a linear extrapolation of
lattice data to the chiral limit. Even failing to accept this perspective, the material
difference between results obtained via the lattice-constrained gap equation and those
found by linear extrapolation of the lattice data must be cause for concern in employing
the latter.
In addition, from a well-defined gap equation it is straightforward to construct
symmetry-preserving Bethe-Salpeter equations whose bound state solutions describe
mesons. We illustrated this via the pion, and calculated its mass and decay constants
in our DSE model of the quenched theory.
Quenched lattice-QCD data on the dressed-quark Schwinger function can be correlated
with dressed-gluon data via a rainbow gap equation so long as that equation’s kernel
possesses enhancement at infrared momenta above that exhibited by the gluon alone.
The required enhancement can be ascribed to a dressing of the quark-gluon vertex. The
solutions of the rainbow gap equation exhibit dynamical chiral symmetry breaking and
are consistent with confinement. The gap equation and related, symmetry-preserving
ladder Bethe-Salpeter equation yield estimates for chiral and physical pion observables
that suggest these quantities are materially underestimated in the quenched theory: |hq̄qi|
by a factor of two and fπ by 30%.
Assuming that existing lattice-QCD data are not afflicted by large systematic errors
associated with finite volume or lattice spacing, we infer from our analysis that quenched
QCD exhibits dynamical chiral symmetry breaking and dressed-quark two-point functions that violate reflection positivity but that chiral and physical pion observables are
91
significantly smaller in the quenched theory than in full QCD.
We have then explored the character of the dressed-quark-gluon vertex and its role
in the gap and Bethe-Salpeter equations.We employed a simple model for the dressedgluon interaction to build an Ansatz for the quark-gluon vertex. The model reduces
coupled integral equations to algebraic equations and thus provides a useful intuitive tool.
We used this framework to argue that data obtained in lattice simulations of quenchedQCD indicate the existence of net attraction in the colour-octet projection of the quarkantiquark scattering kernel.
We observed that the presence of such attraction can affect the uniformity of pointwise
convergence to solutions of the gap and vertex equations. For example, in the timelike
region, the vertex obtained by summing an odd number of loop corrections is pointwise
markedly different from that obtained by summing an even number of loops. The two subseries of vertices so defined follow a different pointwise path to the completely resummed
vertex. This entails that the solutions of two gap equations that are defined via vertex
truncations or vertex Ansätze which appear similar at spacelike momenta need not yield
qualitatively equivalent results for the dressed-quark propagator. This is especially true in
connection with the manifestation of confinement, for which the behaviour of Schwinger
functions at timelike momenta is important.
Our study showed that the dependence of the dressed-quark-gluon vertex on the
current-quark mass is weak until that mass becomes commensurate in magnitude with
the theory’s intrinsic mass-scale. For masses of this magnitude and above, all vertex
dressing is suppressed and the dressed vertex is well approximated by the bare vertex.
It is critical feature of our study that the diagrammatic content of the model we proposed for the vertex is explicitly enumerable because this enables the systematic construction of quark-antiquark and quark-quark scattering kernels that ensure the preservation of
92
all Ward-Takahashi identities associated with strong interaction observables. This guaranteed, in particular, that independent of the number of loop corrections incorporated in the
dressed-quark-gluon vertex, and thereby in the gap equation, the pion was automatically
realised as a Goldstone mode in the chiral limit. Such a result is impossible if one merely
guesses a form for the vertex, no matter how sound the motivation. As a consequence we
could reliably explore the impact on the meson spectrum of attraction in the colour-octet
projection of the quark-antiquark scattering kernel. In accordance with intuition, the mass
of a meson decreases with increasing attraction between the constituents.
We found that the fidelity of an approximate solution for a meson’s mass increases with
the number of loops retained in building the vertex. However, a given consistent truncation
need not yield a solution. This fact is tied to a difference between the convergence paths
followed by the odd-loop vertex series and the even-loop series. In addition, we observed
that with increasing current-quark mass the rainbow-ladder truncation provides an ever
more reliable estimate of the exact vector meson mass; i.e., the mass obtained using
the completely resummed vertex and the completely consistent Bethe-Salpeter equation.
For pseudoscalar mesons, this is even more true because the rainbow-ladder and exact
results are forced by the Ward-Takahashi identity to agree in the chiral limit. This is
practically useful because it means that the parameters of a model meant to be employed
in rainbow-ladder truncation may reliably be fixed by fitting to the values of pseudoscalar
meson quantities calculated in the neighbourhood of the chiral limit. Moreover, both in
rainbow-ladder truncation and with the complete vertex and kernel, the splitting between
pseudoscalar and vector meson masses vanishes as the current-quark mass increases. In
our complete model calculation this splitting is 130 MeV at the c-quark mass and only
40 MeV at the b-quark mass, a pattern which suggests that the pseudoscalar partner of
the Υ(1S) cannot have a mass as low as that currently ascribed to the η b (1S).
93
Finally, we have studied the dressed-quark-gluon vertex via the successful rainbowDSE model developed to analyse the lattice-QCD data for the mass function of the quark
propagator in chapter 3. We use the information provided by pQCD to obtain an Ansatz
for a 1-loop non-perturbative Feynman diagram representation of the vertex. There are
two such diagrams, an Abelian one and a non-Abelian one. The color factor of the nonAbelian diagram indicates a clear domination over the Abelian diagram, which is found
to be the case on evaluating these diagrams. It is also found that the non-Abelian contribution for λ1 is about 6 times more than that suggested by the color factor and greater
than a factor of 1 for the other 2 amplitudes.
Comparison with quenched lattice-data, without any parameter readjustment, indicates excellent results for λ1 and λ3 , but significant differences for λ2 , which appears to
diverge in the lattice calculations as the momentum → 0. Lattice data for λ 2 is difficult
to extract as it mixes with λ1 and so has large error bars. We believe that better lattice
data is required before reaching final conclusions about our comparison for λ 2 .
It is also found that the effects of quenching, after comparing these quenched results
with those of the phenomenlogically successful unquenched Maris-Tandy model are moderate.
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